#8696691

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

4226 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)(3,10)(4,12)(5,7)(6,9)(8,14)(11,15)(13,16)(17,18,22,26)(19,24,20,23)(21,25,30,29)(27,32,28,31)(33,34)(35,39)(36,44)(37,41)(38,42)(40,46)(43,47)(45,48);(1,3,6,5)(2,7,9,10)(4,13,14,11)(8,16,12,15)(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,46)(47,48);(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(33,35)(34,37)(36,43)(38,41)(39,42)(40,47)(44,45)(46,48);(1,4,6,14)(2,8,9,12)(3,11,5,13)(7,15,10,16)(17,20,22,19)(18,24,26,23)(21,27,30,28)(25,31,29,32)(33,35)(34,39)(36,43)(37,42)(38,41)(40,45)(44,47)(46,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,32)(28,31)(33,36)(34,40)(35,43)(37,47)(38,46)(39,45)(41,48)(42,44);(1,5,6,3)(2,7,9,10)(4,11,14,13)(8,16,12,15)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,37,38,39)(34,41,42,35)(36,45,46,47)(40,43,44,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=g,f^2=gh,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, 8603887);