#9379882

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

4325 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)(3,10)(4,5)(6,7)(8,11)(9,15)(12,13)(14,16)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,29)(28,32)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(2,7)(5,9)(8,10)(13,16)(17,19,22,27)(18,23,26,29)(20,21,28,31)(24,30,32,25)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48);(3,11)(8,10)(12,14)(13,16)(19,27)(20,28)(23,29)(24,32)(33,35,37,43)(34,38,40,41)(36,42,45,44)(39,46,48,47);(1,3,6,11)(2,8,7,10)(4,14,15,12)(5,13,9,16)(17,20)(18,24)(19,21)(22,28)(23,30)(25,29)(26,32)(27,31)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,45)(47,48);(1,4)(2,9)(3,12)(5,7)(6,15)(8,13)(10,16)(11,14)(17,21)(18,25)(19,28)(20,27)(22,31)(23,24)(26,30)(29,32)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47);(1,5,6,9)(2,4,7,15)(3,13,11,16)(8,14,10,12)(17,18)(19,29)(20,24)(21,25)(22,26)(23,27)(28,32)(30,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=gh,e^2=h,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, 9287078);