#8696241

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

4311 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,e^2=gh,f^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, 8603437);