#8664427

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

4190 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)(3,9)(4,14)(5,10)(6,7)(8,11)(12,15)(13,16)(17,18,23,20)(19,24,21,25)(22,26,30,28)(27,31,29,32)(33,34,38,39)(35,36,43,40)(37,45,47,41)(42,48,46,44);(1,3,8,6)(2,9,11,7)(4,13,5,12)(10,15,14,16)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(34,39)(36,40)(41,45)(44,48);(1,4,8,5)(2,10,11,14)(3,12,6,13)(7,15,9,16)(17,19,23,21)(18,25,20,24)(22,27,30,29)(26,32,28,31)(33,35)(34,40)(36,39)(37,42)(38,43)(41,48)(44,45)(46,47);(1,5,8,4)(2,10,11,14)(3,13,6,12)(7,15,9,16)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32)(33,36,38,40)(34,35,39,43)(37,44,47,48)(41,42,45,46);(1,6,8,3)(2,9,11,7)(4,13,5,12)(10,16,14,15)(17,21,23,19)(18,24,20,25)(22,29,30,27)(26,31,28,32)(33,37,38,47)(34,41,39,45)(35,42,43,46)(36,44,40,48);(1,7,8,9)(2,6,11,3)(4,15,5,16)(10,13,14,12)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)(33,35)(34,36)(37,46)(38,43)(39,40)(41,48)(42,47)(44,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=h,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, 8571623);