#9159940

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	#55207	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4033 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)(3,7)(4,9)(5,12)(6,8)(10,16)(11,14)(13,15)(18,23)(19,25)(21,29)(28,30)(33,34,39,37)(35,42,40,36)(38,45,46,41)(43,44,47,48);(1,3,6,11)(2,7,8,14)(4,10,15,12)(5,9,16,13)(17,18,22,23)(19,24,25,20)(21,26,29,31)(27,30,32,28)(34,37)(35,40)(41,45)(43,47);(2,8)(3,11)(9,13)(10,12)(17,19,22,25)(18,24,23,20)(21,27,29,32)(26,30,31,28)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,48)(46,47);(1,2)(3,7)(4,13)(5,10)(6,8)(9,15)(11,14)(12,16)(17,19)(18,24)(20,23)(21,27)(22,25)(26,30)(28,31)(29,32)(33,36,39,42)(34,40,37,35)(38,44,46,48)(41,47,45,43);(1,4)(2,9)(3,12)(5,7)(6,15)(8,13)(10,11)(14,16)(17,20,22,24)(18,25,23,19)(21,30,29,28)(26,27,31,32)(33,37,39,34)(35,42,40,36)(38,45,46,41)(43,48,47,44);(1,5)(2,10)(3,9)(4,14)(6,16)(7,15)(8,12)(11,13)(17,21)(18,26)(19,27)(20,28)(22,29)(23,31)(24,30)(25,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=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, 9067136);