#8680008

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

4167 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8587204);