#8662940

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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=g,e^2=g,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, 8570136);