#8963234

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

4118 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8870430);