#9066061

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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4042 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8973257);