#9243388

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

3991 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(34,35)(36,46)(38,47)(41,44);(4,11)(8,14)(10,15)(12,16)(17,18,22,19)(20,25,27,23)(21,26,29,24)(28,30,32,31)(33,34,39,35)(36,40,46,43)(37,44,48,41)(38,45,47,42);(2,7)(3,9)(8,14)(10,15)(18,19)(20,27)(24,26)(28,32)(33,35,39,34)(36,40,46,43)(37,41,48,44)(38,45,47,42);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,36)(34,40)(35,43)(37,47)(38,48)(39,46)(41,45)(42,44);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,37,39,48)(34,41,35,44)(36,47,46,38)(40,45,43,42);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,47)(34,42,35,45)(36,48,46,37)(40,44,43,41)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^2=gh,e^2=g,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, 9150584);