#8873646

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

4087 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8780842);