#8664430

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

4162 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=gh,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, 8571626);