#7658760

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

4724 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

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, 7565956);