#8621657

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

4145 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,7)(3,11,12,8)(4,5,10,9)(13,14,16,15)(17,18,22,25)(19,26,21,23)(20,24,29,32)(27,31,28,30)(33,34,38,40)(35,39,43,44)(36,41,46,48)(37,42,47,45);(2,7)(3,12)(4,10)(14,15)(17,19,22,21)(18,23,25,26)(20,28,29,27)(24,31,32,30)(33,35)(34,39)(36,45)(37,41)(38,43)(40,44)(42,46)(47,48);(2,7)(4,10)(8,11)(13,16)(20,29)(24,32)(27,28)(30,31)(34,40)(35,43)(41,48)(42,45);(1,3,6,12)(2,8,7,11)(4,13,10,16)(5,15,9,14)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,36,38,46)(34,41,40,48)(35,42,43,45)(37,39,47,44);(1,4,6,10)(2,9,7,5)(3,13,12,16)(8,14,11,15)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,32)(26,31)(33,34)(35,44)(36,41)(37,42)(38,40)(39,43)(45,47)(46,48);(1,5,6,9)(2,10,7,4)(3,14,12,15)(8,13,11,16)(17,21,22,19)(18,23,25,26)(20,28,29,27)(24,30,32,31)(33,37)(34,42)(35,41)(36,44)(38,47)(39,46)(40,45)(43,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,d^2=h,e^2=hi,f^2=g,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, 8528853);