#8505075

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

4558 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)(3,5)(4,7)(6,8)(9,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,24)(22,23)(25,26,31,28)(27,32,37,35)(29,38,40,33)(30,34,36,39);(2,3)(6,7)(11,20)(12,22)(15,16)(17,23)(25,27)(26,32)(28,35)(29,30)(31,37)(33,39)(34,38)(36,40);(1,2,5,3)(4,6,8,7)(9,11)(10,15)(12,21)(13,17)(14,20)(16,19)(18,22)(23,24)(25,26,31,28)(27,35,37,32)(29,33,40,38)(30,34,36,39);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,21)(13,23)(14,20)(15,19)(17,24)(18,22)(25,28,31,26)(27,35,37,32)(29,38,40,33)(30,34,36,39);(1,4,5,8)(2,6,3,7)(9,12)(10,17)(11,21)(13,15)(14,22)(16,24)(18,20)(19,23)(25,29)(26,33)(27,36)(28,38)(30,37)(31,40)(32,34)(35,39);(9,13,14,24)(10,18,19,21)(11,17,20,23)(12,15,22,16)(25,30,31,36)(26,34,28,39)(27,29,37,40)(32,38,35,33)

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=ghi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8412271);