#8692567

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	#36905	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4323 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8599763);