#8196919

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

4540 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=gi,e^2=h,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, 8104115);