#9232773

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

3885 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,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, 9139969);