#8455681

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

4397 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)(3,4)(5,7)(6,8)(10,16)(11,12)(17,23)(18,20)(26,30)(31,37)(32,38)(39,40);(1,3)(2,6)(4,7)(5,8)(9,10,15,16)(11,14,12,19)(13,17,22,23)(18,21,20,24)(26,30)(27,33)(32,38)(34,35);(9,11,15,12)(10,14,16,19)(13,18,22,20)(17,21,23,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,40)(36,38);(2,7)(4,6)(9,12,15,11)(10,14,16,19)(13,20,22,18)(17,21,23,24)(25,27,29,33)(26,31,30,37)(28,35,36,34)(32,40,38,39);(1,3)(2,6)(4,7)(5,8)(9,13,15,22)(10,17,16,23)(11,18,12,20)(14,21,19,24);(1,4,5,6)(2,3,7,8)(9,14)(10,12)(11,16)(13,21)(15,19)(17,20)(18,23)(22,24)(25,28,29,36)(26,32,30,38)(27,34,33,35)(31,39,37,40)

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,d^2=h,e^2=hi,f^2=gh,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, 8362877);