#8926711

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5328 conjugacy classes of subgroups; only the 2870 normal subgroups are shown.

normal subgroups only

This group has 5328 conjugacy classes of subgroups — too many to draw. Only its 2870 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(2,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18,24,27)(19,28,20,22)(21,25,35,40)(23,26,38,42)(29,33,31,36)(30,41,32,37)(34,39,46,48)(43,45,44,47);(3,7)(6,8)(10,14)(11,16)(17,19,24,20)(18,22,27,28)(21,29,35,31)(23,30,38,32)(25,36,40,33)(26,37,42,41)(34,43,46,44)(39,47,48,45);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,20,24,19)(18,22,27,28)(21,31,35,29)(23,30,38,32)(25,36,40,33)(26,41,42,37)(34,43,46,44)(39,45,48,47);(9,12)(10,11)(13,15)(14,16)(17,21)(18,25)(19,29)(20,31)(22,36)(23,34)(24,35)(26,39)(27,40)(28,33)(30,43)(32,44)(37,47)(38,46)(41,45)(42,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,14)(15,16)(17,22)(18,19)(20,27)(21,33)(23,37)(24,28)(25,31)(26,30)(29,40)(32,42)(34,45)(35,36)(38,41)(39,44)(43,48)(46,47);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,23)(18,26)(19,30)(20,32)(21,34)(22,37)(24,38)(25,39)(27,42)(28,41)(29,43)(31,44)(33,45)(35,46)(36,47)(40,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72227972, #72268093
Fitting subgroup of	#72227713, #72227717
maximal subgroup of	#72227296, #72227298, #72227712, …
maximal normal subgroup of	#72227296, #72227298, #72227712, …
maximal quotient of	#72267914, #72268079, #72268312

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833907);