#8701395

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

4365 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=ghi,f^2=gi,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, 8608591);