#9222115

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	#47087	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4195 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)(3,8)(4,6)(5,9)(7,11)(10,16)(12,14)(13,15)(17,18,21,25)(19,24,27,22)(20,28,30,23)(26,31,29,32)(33,34,39,44)(35,42,37,40)(36,46,47,41)(38,43,45,48);(1,3)(2,7)(4,8)(5,10)(6,11)(9,15)(12,16)(13,14)(17,18)(19,22)(20,23)(21,25)(24,27)(26,31)(28,30)(29,32)(33,35,39,37)(34,40,44,42)(36,38,47,45)(41,43,46,48);(17,19)(18,22)(20,26)(21,27)(23,31)(24,25)(28,32)(29,30)(33,36,39,47)(34,41,44,46)(35,45,37,38)(40,48,42,43);(1,4,6,2)(3,8,11,7)(5,9,14,12)(10,15,13,16)(17,20,21,30)(18,23,25,28)(19,26,27,29)(22,31,24,32)(33,37,39,35)(34,42,44,40)(36,38,47,45)(41,43,46,48);(1,3)(2,8)(4,7)(5,13)(6,11)(9,15)(10,14)(12,16)(17,19)(18,24)(20,29)(21,27)(22,25)(23,31)(26,30)(28,32)(33,38)(34,43)(35,36)(37,47)(39,45)(40,41)(42,46)(44,48);(1,5,6,14)(2,9,4,12)(3,10,11,13)(7,15,8,16)(17,18)(19,22)(20,28)(21,25)(23,30)(24,27)(26,32)(29,31)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=g,f^2=gh,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, 9129311);