#9017376

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
prime radical	#44249	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3993 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=hi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8924572);