#9379895

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4407 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)(3,5)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,29)(28,32)(33,34,38,39)(35,40,46,41)(36,37,43,42)(44,45,47,48);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(17,19,22,27)(18,23,26,29)(20,21,28,31)(24,30,32,25)(34,39)(37,42)(40,41)(45,48);(2,4)(3,10)(9,12)(11,16)(19,27)(20,28)(23,29)(24,32)(33,35,38,46)(34,40,39,41)(36,44,43,47)(37,45,42,48);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,20)(18,24)(19,21)(22,28)(23,30)(25,29)(26,32)(27,31)(33,35)(34,41)(36,47)(37,45)(38,46)(39,40)(42,48)(43,44);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,21)(18,25)(19,28)(20,27)(22,31)(23,24)(26,30)(29,32)(33,36,38,43)(34,42,39,37)(35,44,46,47)(40,48,41,45);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,18)(19,29)(20,24)(21,25)(22,26)(23,27)(28,32)(30,31)(33,37,38,42)(34,43,39,36)(35,45,46,48)(40,47,41,44)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9287091);