#9333326

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

4329 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)(3,5)(4,9)(6,7)(8,11)(10,13)(12,14)(15,16)(17,18,23,20)(19,24,26,21)(22,25,31,28)(27,32,30,29)(34,38)(36,44)(39,41)(42,48);(2,7)(5,8)(9,14)(13,15)(17,19,23,26)(18,24,20,21)(22,30,31,27)(25,29,28,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(4,12)(9,14)(10,16)(13,15)(19,26)(21,24)(27,30)(29,32)(33,34,37,38)(35,39,43,41)(36,40,44,45)(42,46,48,47);(1,3)(2,8)(4,10)(5,7)(6,11)(9,15)(12,16)(13,14)(17,20,23,18)(19,21,26,24)(22,28,31,25)(27,29,30,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,4,6,12)(2,9,7,14)(3,10,11,16)(5,13,8,15)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,30)(31,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5,6,8)(2,3,7,11)(4,13,12,15)(9,10,14,16)(17,22,23,31)(18,25,20,28)(19,27,26,30)(21,29,24,32)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=hi,e^2=gh,f^2=gi,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, 9240522);