#8696292

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

4572 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gi,d^2=h,e^2=h,f^2=g,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, 8603488);