#8931035

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

4224 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)(3,5)(4,8)(6,7)(11,17)(15,22)(18,20)(23,24)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(2,7)(3,8)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(29,37)(32,36)(34,40)(38,39);(1,3,6,8)(2,4,7,5)(10,14)(11,17)(16,19)(18,20)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,4,6,5)(2,3,7,8)(9,11)(10,15)(12,20)(13,17)(14,22)(16,24)(18,21)(19,23)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(1,2)(3,4)(5,8)(6,7)(9,12,13,21)(10,16,14,19)(11,18,17,20)(15,23,22,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40);(1,5,6,4)(2,8,7,3)(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40)

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8838231);