#9009224

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	#57271	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4062 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=i,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, 8916420);