#9243601

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

4131 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,23,21)(19,24,27,30)(20,25,28,29)(22,31,32,26)(33,34,39,40)(35,41,36,45)(37,38,42,48)(43,47,46,44);(3,9)(7,13)(10,15)(14,16)(17,19,23,27)(18,24,21,30)(20,22,28,32)(25,31,29,26)(34,40)(38,48)(41,45)(44,47);(2,6)(7,13)(8,11)(14,16)(18,21)(19,27)(20,28)(26,31)(33,35,39,36)(34,41,40,45)(37,46,42,43)(38,44,48,47);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20)(18,25)(19,22)(21,29)(23,28)(24,31)(26,30)(27,32)(33,36,39,35)(34,41,40,45)(37,46,42,43)(38,47,48,44);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,21,23,18)(19,24,27,30)(20,29,28,25)(22,31,32,26)(33,37,39,42)(34,38,40,48)(35,43,36,46)(41,47,45,44);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22)(18,26)(19,20)(21,31)(23,32)(24,29)(25,30)(27,28)(33,38)(34,42)(35,44)(36,47)(37,40)(39,48)(41,43)(45,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=g,d^2=g,e^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, 9150797);