#8680011

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

4166 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=gi,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, 8587207);