#9236445

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3992 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)(4,15)(5,10)(6,14)(7,11)(9,13)(12,16)(17,18,23,21)(19,27,26,22)(20,25,30,24)(28,29,32,31)(33,34)(35,44)(36,41)(37,40)(38,42)(39,43)(45,48)(46,47);(1,3)(2,8)(4,5)(6,11)(7,14)(9,16)(10,15)(12,13)(17,19,23,26)(18,22,21,27)(20,28,30,32)(24,29,25,31)(33,35,39,37)(34,40,43,44)(36,42,46,45)(38,41,48,47);(1,4)(2,9)(3,12)(5,14)(6,16)(7,13)(8,10)(11,15)(17,20)(18,24)(19,28)(21,25)(22,29)(23,30)(26,32)(27,31)(33,36,39,46)(34,41,43,47)(35,42,37,45)(38,40,48,44);(1,4)(2,9)(3,5)(6,10)(7,13)(8,16)(11,15)(12,14)(17,20)(18,25)(19,28)(21,24)(22,31)(23,30)(26,32)(27,29)(33,34)(35,40)(36,41)(37,44)(38,45)(39,43)(42,48)(46,47);(1,5)(2,10)(3,13)(4,14)(6,15)(7,12)(8,9)(11,16)(17,21,23,18)(19,22,26,27)(20,25,30,24)(28,29,32,31)(33,37,39,35)(34,40,43,44)(36,45,46,42)(38,41,48,47);(1,6,7,8)(2,3,11,14)(4,16,13,10)(5,9,12,15)(17,22)(18,26)(19,21)(20,29)(23,27)(24,32)(25,28)(30,31)(33,38,39,48)(34,42,43,45)(35,41,37,47)(36,40,46,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9143641);