#8656235

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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=gi,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8563431);