#8550933

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

4122 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=g,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8458129);