#8696055

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

4018 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)(17,18,24,26)(19,27,23,22)(20,25,21,31)(28,32,30,29)(33,34)(35,42)(36,39)(37,44)(38,41)(40,46)(43,45)(47,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,15)(12,16)(13,14)(17,19,24,23)(18,22,26,27)(20,28,21,30)(25,29,31,32)(34,36)(35,41)(40,44)(43,47);(1,3,6,4)(2,7,10,8)(5,11,14,12)(9,15,13,16)(17,20,24,21)(18,25,26,31)(19,28,23,30)(22,29,27,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,4,6,3)(2,8,10,7)(5,11,14,12)(9,15,13,16)(17,21,24,20)(18,25,26,31)(19,28,23,30)(22,32,27,29)(33,36,39,34)(35,42,41,38)(37,44,46,40)(43,48,47,45);(1,2)(3,7)(4,8)(5,13)(6,10)(9,14)(11,16)(12,15)(17,22,24,27)(18,19,26,23)(20,29,21,32)(25,28,31,30)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48);(1,5,6,14)(2,9,10,13)(3,11,4,12)(7,15,8,16)(17,23,24,19)(18,22,26,27)(20,30,21,28)(25,29,31,32)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=g,e^2=gh,f^2=g,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, 8603251);