#8550934

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

4101 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:

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

Polycyclic presentation:

2^9|a^2=ghi,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, 8458130);