#9240913

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

4145 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)(3,10,9,5)(4,8,12,14)(11,16,15,13)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(33,34)(35,37)(36,39)(38,44)(40,42)(41,46)(43,47)(45,48);(2,7)(4,12)(5,10)(11,15)(17,18,22,19)(20,25,27,23)(21,26,29,24)(28,30,32,31)(33,35)(34,40)(36,37)(38,43)(39,42)(41,47)(44,45)(46,48);(2,7)(3,9)(4,12)(13,16)(18,19)(20,27)(24,26)(28,32)(34,36)(35,42)(41,44)(43,48);(1,3,6,9)(2,5,7,10)(4,11,12,15)(8,13,14,16)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,36,39,34)(35,37,42,40)(38,44,46,41)(43,45,48,47);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,37)(34,35)(36,42)(38,45)(39,40)(41,43)(44,48)(46,47);(1,5)(2,9)(3,7)(4,13)(6,10)(8,15)(11,14)(12,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=h,e^2=gh,f^2=gh,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, 9148109);