#8455935

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

4172 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8363131);