#8634821

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8542017);