#8942429

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

4211 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=h,e^2=gh,f^2=gh,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, 8849625);