#9224285

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	#43662	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4110 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9131481);