#8701264

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

4590 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)(3,5)(4,7)(6,8)(9,10)(11,15)(12,20)(13,17)(14,18)(16,22)(19,23)(21,24);(2,3)(6,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,22)(23,24)(25,26)(27,33)(28,29)(30,34)(31,37)(32,38)(35,39)(36,40);(1,2,5,3)(4,6,8,7)(9,11)(10,13)(12,19)(14,17)(15,18)(16,23)(20,21)(22,24)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,19)(13,18)(14,17)(16,21)(20,23)(22,24)(25,28,30,38)(26,32,34,29)(27,35,37,40)(31,39,33,36);(1,4)(2,6)(3,7)(5,8)(9,12,14,22)(10,16,18,20)(11,19,17,24)(13,23,15,21)(25,27)(26,33)(28,35)(29,39)(30,37)(31,34)(32,36)(38,40);(9,13,14,15)(10,17,18,11)(12,21,22,23)(16,19,20,24)(25,29,30,32)(26,28,34,38)(27,36,37,39)(31,35,33,40)

Polycyclic presentation:

2^9|c^2=g,d^2=ghi,e^2=gh,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, 8608460);