#8108421

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

4614 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:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

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, 8015617);