#8696063

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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8603259);