#8533171

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

4328 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=i,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8440367);