#9009247

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3923 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=gi,e^2=gi,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, 8916443);