#8690953

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

4587 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8598149);