#8806795

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

4143 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=ghi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8713991);