#8696774

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

4202 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)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(18,22)(19,25)(24,27)(26,28)(33,34,40,39)(35,38,44,43)(36,45,37,41)(42,48,46,47);(2,3)(7,9)(8,10)(15,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,35)(34,38)(36,46)(37,42)(39,43)(40,44)(41,48)(45,47);(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,36,40,37)(34,41,39,45)(35,42,44,46)(38,47,43,48);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,37,40,36)(34,41,39,45)(35,42,44,46)(38,48,43,47);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,19,21,25)(18,23,22,30)(20,28,29,26)(24,32,27,31)(33,38)(34,35)(36,47)(37,48)(39,44)(40,43)(41,42)(45,46);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,39,40,34)(35,43,44,38)(36,45,37,41)(42,48,46,47)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=gh,e^2=g,f^2=gh,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, 8603970);