#9017388

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4091 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=ghi,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, 8924584);