#8334382

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

4060 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8241578);