#8750257

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

4425 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8657453);