#8467241

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

4446 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)(3,7)(4,6)(5,8)(10,16)(11,12)(17,23)(18,20)(25,26,31,30)(27,32,37,35)(28,33,38,40)(29,39,36,34);(1,3)(2,5)(4,8)(6,7)(9,10,15,16)(11,14,12,19)(13,17,22,23)(18,21,20,24)(25,26,31,30)(27,35,37,32)(28,33,38,40)(29,34,36,39);(9,11,15,12)(10,14,16,19)(13,18,22,20)(17,21,23,24)(25,27,31,37)(26,32,30,35)(28,36,38,29)(33,34,40,39);(2,6)(5,7)(9,12,15,11)(10,14,16,19)(13,20,22,18)(17,21,23,24)(25,28)(26,33)(27,36)(29,37)(30,40)(31,38)(32,34)(35,39);(1,3)(2,5)(4,8)(6,7)(9,13,15,22)(10,17,16,23)(11,18,12,20)(14,21,19,24)(25,29,31,36)(26,34,30,39)(27,28,37,38)(32,40,35,33);(1,2)(3,5)(4,6)(7,8)(9,14)(10,12)(11,16)(13,21)(15,19)(17,20)(18,23)(22,24)(25,30,31,26)(27,35,37,32)(28,33,38,40)(29,39,36,34)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,d^2=h,e^2=hi,f^2=i,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, 8374437);