#8772761

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

4646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8679957);