#8700388

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

4173 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8607584);