#9267000

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

4003 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=i,d^2=gi,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, 9174196);