#9243322

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

4374 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gh,d^2=g,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, 9150518);