#9290497

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) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9197693);