#9021948

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	#53440	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4112 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=g,f^2=i,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, 8929144);