#8630896

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

4195 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)(3,9)(4,10)(5,8)(6,7)(11,15)(12,16)(13,14)(17,18,22,25)(19,21,24,31)(20,27,30,23)(26,32,28,29)(33,34,39,42)(35,43,41,37)(36,46,38,40)(44,48,45,47);(2,7)(3,10)(8,14)(11,16)(17,19,22,24)(18,21,25,31)(20,28,30,26)(23,32,27,29)(33,35,39,41)(34,37,42,43)(36,44,38,45)(40,48,46,47);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,20,22,30)(18,23,25,27)(19,26,24,28)(21,29,31,32)(33,36,39,38)(34,40,42,46)(35,44,41,45)(37,48,43,47);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,19)(18,21)(20,26)(22,24)(23,29)(25,31)(27,32)(28,30)(33,34)(35,43)(36,40)(37,41)(38,46)(39,42)(44,47)(45,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,27)(31,32)(33,37,39,43)(34,41,42,35)(36,47,38,48)(40,44,46,45);(1,5,6,13)(2,8,7,14)(3,11,10,16)(4,12,9,15)(17,21)(18,24)(19,25)(20,29)(22,31)(23,28)(26,27)(30,32)(33,38,39,36)(34,40,42,46)(35,45,41,44)(37,48,43,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8538092);