#9333336

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:

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

Polycyclic presentation:

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

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, 9240532);