#9232831

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3865 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=i,e^2=gh,f^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, 9140027);