#9026957

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

4225 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)(4,8,12,16)(11,14,13,15)(17,18,23,20)(19,24,27,28)(21,25,31,29)(22,30,32,26)(33,34,38,36)(35,41,43,39)(37,44,46,40)(42,47,45,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,15)(12,16)(13,14)(17,19)(18,24)(20,28)(21,30)(22,29)(23,27)(25,32)(26,31)(34,36)(39,41)(40,44)(47,48);(3,9)(4,12)(5,7)(8,16)(18,20)(19,27)(21,31)(22,32)(33,35,38,43)(34,39,36,41)(37,45,46,42)(40,48,44,47);(1,3,6,9)(2,7,10,5)(4,13,12,11)(8,14,16,15)(17,20,23,18)(19,28,27,24)(21,25,31,29)(22,30,32,26)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46);(1,4,6,12)(2,8,10,16)(3,11,9,13)(5,14,7,15)(17,21)(18,25)(19,26)(20,29)(22,24)(23,31)(27,30)(28,32)(33,36,38,34)(35,39,43,41)(37,44,46,40)(42,47,45,48);(1,5,6,7)(2,9,10,3)(4,14,12,15)(8,13,16,11)(17,22,23,32)(18,26,20,30)(19,29,27,25)(21,24,31,28)(33,37,38,46)(34,40,36,44)(35,42,43,45)(39,47,41,48)

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8934153);