#8689263

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

4378 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,e^2=g,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, 8596459);