#8467161

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

4443 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=i,d^2=gh,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8374357);