#9240904

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

4282 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)(3,9)(4,8)(5,10)(6,7)(11,15)(12,14)(13,16)(18,19)(23,25)(24,26)(30,31)(33,34,38,35)(36,39,43,41)(37,42,45,40)(44,47,48,46);(2,7)(4,12)(5,9)(11,16)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(33,34,38,35)(36,41,43,39)(37,42,45,40)(44,46,48,47);(2,7)(3,10)(4,12)(13,15)(20,27)(23,25)(28,32)(30,31)(34,35)(36,43)(40,42)(44,48);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,16)(14,15)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,35,38,34)(36,41,43,39)(37,42,45,40)(44,47,48,46);(1,4,6,12)(2,8,7,14)(3,11,10,16)(5,13,9,15)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,36,38,43)(34,39,35,41)(37,44,45,48)(40,46,42,47);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,37,38,45)(34,40,35,42)(36,44,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=gh,e^2=h,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, 9148100);