#8664435

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

4097 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)(3,9)(4,14)(5,10)(6,7)(8,11)(12,15)(13,16)(17,18)(19,29)(20,26)(21,23)(22,27)(24,28)(25,32)(30,31)(33,34,38,42)(35,41,46,39)(36,37,47,40)(43,48,44,45);(1,3,8,6)(2,9,11,7)(4,13,5,12)(10,15,14,16)(17,19,24,32)(18,25,28,29)(20,30,21,22)(23,31,26,27)(33,35)(34,39)(36,43)(37,45)(38,46)(40,48)(41,42)(44,47);(1,4,8,5)(2,10,11,14)(3,12,6,13)(7,15,9,16)(17,20,24,21)(18,26,28,23)(19,30,32,22)(25,27,29,31)(33,36)(34,40)(35,43)(37,42)(38,47)(39,48)(41,45)(44,46);(1,5,8,4)(2,10,11,14)(3,13,6,12)(7,15,9,16)(17,21,24,20)(18,26,28,23)(19,22,32,30)(25,27,29,31)(33,36)(34,40)(35,44)(37,42)(38,47)(39,45)(41,48)(43,46);(1,6,8,3)(2,9,11,7)(4,13,5,12)(10,16,14,15)(17,22)(18,27)(19,20)(21,32)(23,25)(24,30)(26,29)(28,31)(33,37,38,40)(34,36,42,47)(35,45,46,48)(39,43,41,44);(1,7,8,9)(2,6,11,3)(4,15,5,16)(10,13,14,12)(17,23,24,26)(18,21,28,20)(19,31,32,27)(22,29,30,25)(33,35)(34,41)(36,43)(37,48)(38,46)(39,42)(40,45)(44,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8571631);