#8712364

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

4210 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=h,f^2=g,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, 8619560);