#10464547

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

4163 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10371743);