#8466921

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

4491 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)(3,6)(4,7)(5,8)(10,14)(11,17)(16,19)(18,23)(25,26,31,30)(27,32,37,35)(28,33,38,40)(29,39,36,34);(1,3,5,7)(2,6,8,4)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,26,31,30)(27,35,37,32)(28,33,38,40)(29,34,36,39);(1,3)(2,4)(5,7)(6,8)(25,27,31,37)(26,32,30,35)(28,36,38,29)(33,34,40,39);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,28)(26,33)(27,36)(29,37)(30,40)(31,38)(32,34)(35,39);(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,29,31,36)(26,34,30,39)(27,28,37,38)(32,40,35,33);(1,4,5,6)(2,7,8,3)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(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=i,c^2=hi,e^2=ghi,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, 8374117);