#8664428

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

4188 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)(35,41)(36,38)(37,44)(39,43)(40,46)(42,47)(45,48);(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)(34,36)(39,41)(40,44)(47,48);(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)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46);(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)(33,36,38,34)(35,41,43,39)(37,44,46,40)(42,48,45,47);(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)(33,37,38,46)(34,40,36,44)(35,42,43,45)(39,47,41,48);(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)(34,41)(36,39)(37,45)(38,43)(40,47)(42,46)(44,48)

Polycyclic presentation:

2^9|a^2=ghi,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, 8571624);