#8718367

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

4581 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^2=gi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8625563);