#8630922

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

4304 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)(3,8)(4,10)(5,9)(6,7)(11,15)(12,16)(13,14)(17,18,24,30)(19,28,22,25)(20,27,21,26)(23,29,32,31)(35,41)(38,42)(43,47)(46,48);(2,7)(8,10)(9,14)(15,16)(17,19)(18,25)(20,29)(21,31)(22,24)(23,27)(26,32)(28,30)(33,34,37,40)(35,42,41,38)(36,39,45,44)(43,48,47,46);(1,3,6,4)(2,8,7,10)(5,11,13,12)(9,15,14,16)(17,20,24,21)(18,26,30,27)(19,29,22,31)(23,25,32,28)(33,35,37,41)(34,38,40,42)(36,43,45,47)(39,46,44,48);(1,4,6,3)(2,8,7,10)(5,11,13,12)(9,16,14,15)(17,21,24,20)(18,27,30,26)(19,29,22,31)(23,25,32,28)(33,34)(35,38)(36,44)(37,40)(39,45)(41,42)(43,48)(46,47);(1,2)(3,10)(4,8)(5,9)(6,7)(11,16)(12,15)(13,14)(17,22,24,19)(18,28,30,25)(20,29,21,31)(23,26,32,27)(33,35)(34,38)(36,43)(37,41)(39,46)(40,42)(44,48)(45,47);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,23)(18,29)(19,27)(20,25)(21,28)(22,26)(24,32)(30,31)(33,36)(34,39)(35,43)(37,45)(38,46)(40,44)(41,47)(42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gi,d^2=g,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8538118);