#8841730

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

4269 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=hi,e^2=g,f^2=g,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=dgi,e^d=egi

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, 8748926);