#9015561

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	#43367	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4162 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=ghi,d^2=hi,e^2=i,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, 8922757);