#8665392

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

4636 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8572588);