#9240910

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

4297 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)(3,5)(4,9)(6,7)(8,11)(10,13)(12,14)(15,16)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(33,34)(35,41)(36,38)(37,40)(39,43)(42,48)(44,46)(45,47);(2,7)(5,8)(9,14)(13,15)(17,18,22,19)(20,25,27,23)(21,26,29,24)(28,30,32,31)(33,35,38,43)(34,39,36,41)(37,45,46,42)(40,48,44,47);(4,12)(9,14)(10,16)(13,15)(18,19)(20,27)(24,26)(28,32)(34,36)(37,46)(39,41)(42,45);(1,3)(2,8)(4,10)(5,7)(6,11)(9,15)(12,16)(13,14)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,36,38,34)(35,41,43,39)(37,44,46,40)(42,48,45,47);(1,4,6,12)(2,9,7,14)(3,10,11,16)(5,13,8,15)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45);(1,5,6,8)(2,3,7,11)(4,13,12,15)(9,10,14,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=h,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, 9148106);