#8850694

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

4293 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=ghi,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, 8757890);