#8696655

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

4217 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,e^2=g,f^2=h,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, 8603851);