#10464577

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

4149 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)(19,20)(21,24)(22,30)(23,26)(25,31)(27,29)(28,32)(33,34)(35,36)(37,41)(38,42)(39,40)(43,44)(45,48)(46,47);(3,10)(6,9)(11,12)(15,16)(17,19,23,24)(18,21,26,20)(22,27,31,32)(25,29,30,28)(34,40)(36,41)(42,47)(44,48);(1,3)(2,6)(4,11)(5,10)(7,15)(8,9)(12,14)(13,16)(17,20,23,21)(18,24,26,19)(22,28,31,29)(25,32,30,27)(33,35,39,37)(34,36,40,41)(38,43,46,45)(42,44,47,48);(1,3)(2,6)(4,12)(5,10)(7,16)(8,9)(11,14)(13,15)(17,21,23,20)(18,24,26,19)(22,29,31,28)(25,32,30,27)(33,36,39,41)(34,35,40,37)(38,44,46,48)(42,43,47,45);(1,4,5,14)(2,7,8,13)(3,11,10,12)(6,15,9,16)(17,18)(19,21)(20,24)(22,30)(23,26)(25,31)(27,28)(29,32)(33,37,39,35)(34,41,40,36)(38,43,46,45)(42,44,47,48);(1,2)(3,6)(4,13)(5,8)(7,14)(9,10)(11,16)(12,15)(17,22,23,31)(18,25,26,30)(19,27,24,32)(20,28,21,29)(33,38,39,46)(34,42,40,47)(35,43,37,45)(36,44,41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=g,e^2=gh,f^2=g,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, 10371773);