#8648953

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

4472 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)(4,6)(5,8)(9,10,15,12)(11,16,19,13)(14,17,22,20)(18,23,24,21)(25,26,32,38)(27,39,40,33)(28,35,29,34)(30,31,37,36);(1,2,4,6)(3,5,8,7)(9,11)(10,16)(12,13)(14,18)(15,19)(17,23)(20,21)(22,24)(25,27)(26,33)(28,36)(29,31)(30,34)(32,40)(35,37)(38,39);(1,2)(3,5)(4,6)(7,8)(9,10,15,12)(11,13,19,16)(14,17,22,20)(18,21,24,23)(25,28,32,29)(26,34,38,35)(27,36,40,31)(30,39,37,33);(1,3,4,8)(2,5,6,7)(9,12,15,10)(11,13,19,16)(14,17,22,20)(18,23,24,21)(25,29,32,28)(26,35,38,34)(27,36,40,31)(30,39,37,33);(9,13)(10,11)(12,19)(14,21)(15,16)(17,18)(20,24)(22,23)(25,30,32,37)(26,36,38,31)(27,35,40,34)(28,33,29,39);(9,14,15,22)(10,17,12,20)(11,18,19,24)(13,21,16,23)(25,31,32,36)(26,37,38,30)(27,29,40,28)(33,35,39,34)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8556149);