#9243146

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

4097 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)(3,5,9,13)(4,8,11,14)(10,12,15,16)(17,18)(19,22)(20,27)(21,23)(24,29)(25,31)(26,32)(28,30)(33,34)(35,41)(36,44)(37,38)(39,40)(42,46)(43,45)(47,48);(4,11)(8,14)(10,15)(12,16)(17,19,21,25)(18,22,23,31)(20,28,29,26)(24,32,27,30)(34,40)(38,48)(41,44)(43,46);(2,7)(3,9)(8,14)(10,15)(18,23)(19,25)(24,27)(26,28)(33,35,39,36)(34,41,40,44)(37,45,47,42)(38,43,48,46);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,18)(19,22)(20,24)(21,23)(25,31)(26,30)(27,29)(28,32)(33,36,39,35)(34,41,40,44)(37,45,47,42)(38,46,48,43);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,19)(18,22)(20,26)(21,25)(23,31)(24,30)(27,32)(28,29)(33,37)(34,38)(35,42)(36,45)(39,47)(40,48)(41,46)(43,44);(1,5)(2,3)(4,12)(6,13)(7,9)(8,10)(11,16)(14,15)(17,20)(18,24)(19,26)(21,29)(22,30)(23,27)(25,28)(31,32)(33,38)(34,37)(35,43)(36,46)(39,48)(40,47)(41,45)(42,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^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, 9150342);