#7642105

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5154 conjugacy classes of subgroups; only the 3194 normal subgroups are shown.

normal subgroups only

This group has 5154 conjugacy classes of subgroups — too many to draw. Only its 3194 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)(3,7)(9,10,12,14)(11,15,16,13)(21,33)(26,35)(29,43)(32,44)(37,40)(38,45)(42,48)(46,47);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(18,23)(20,31)(24,27)(26,35)(28,41)(32,44)(37,40)(42,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18,22,23)(19,27,30,24)(20,25,31,34)(21,26,33,35)(28,39,41,36)(29,40,43,37)(32,38,44,45)(42,47,48,46);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,19,22,30)(18,24,23,27)(20,28,31,41)(21,29,33,43)(25,36,34,39)(26,37,35,40)(32,42,44,48)(38,46,45,47);(17,20)(18,25)(19,28)(21,32)(22,31)(23,34)(24,36)(26,38)(27,39)(29,42)(30,41)(33,44)(35,45)(37,46)(40,47)(43,48);(17,21)(18,26)(19,29)(20,32)(22,33)(23,35)(24,37)(25,38)(27,40)(28,42)(30,43)(31,44)(34,45)(36,46)(39,47)(41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=ehi,d^c=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72267532, #72267533, #72267890, …
Fitting subgroup of	#72267532
maximal subgroup of	#72267531, #72267532, #72267887
maximal normal subgroup of	#72267531, #72267532, #72267887
maximal quotient of	#72267545, #72267604, #72267607

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549301);