#8416908

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

5276 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324104);