#8393762

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

4495 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300958);