#8724823

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

4442 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8632019);