#8630891

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

4018 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)(3,10)(4,12)(5,7)(6,8)(9,14)(11,15)(13,16)(17,18,22,25)(19,21,24,31)(20,27,30,23)(26,32,28,29)(33,34,38,39)(35,37,42,40)(36,44,46,41)(43,47,48,45);(1,3,7,6)(2,8,5,10)(4,13,14,11)(9,16,12,15)(17,19,22,24)(18,21,25,31)(20,28,30,26)(23,32,27,29)(34,39)(35,42)(41,44)(43,48);(2,5)(8,10)(9,12)(15,16)(17,20,22,30)(18,23,25,27)(19,26,24,28)(21,29,31,32)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,4)(2,9)(3,11)(5,12)(6,13)(7,14)(8,15)(10,16)(17,19)(18,21)(20,26)(22,24)(23,29)(25,31)(27,32)(28,30)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,5,7,2)(3,8,6,10)(4,12,14,9)(11,15,13,16)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,27)(31,32)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,6,7,3)(2,8,5,10)(4,11,14,13)(9,16,12,15)(17,21)(18,24)(19,25)(20,29)(22,31)(23,28)(26,27)(30,32)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=i,e^2=hi,f^2=g,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, 8538087);