#9333338

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

4334 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=ghi,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, 9240534);