#8197307

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

4488 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=gh,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104503);