#8773034

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

4067 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=h,e^2=gh,f^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=dgi,e^c=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, 8680230);