#8773007

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

4268 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)(3,5)(4,8)(6,7)(9,10,15,12)(11,18,14,16)(13,17,22,20)(19,24,21,23)(25,26,30,33)(27,29,32,38)(28,31,37,40)(34,36,35,39);(2,7)(3,8)(9,10,15,12)(11,16,14,18)(13,20,22,17)(19,24,21,23)(27,32)(29,38)(34,35)(36,39);(1,3,6,8)(2,4,7,5)(9,11,15,14)(10,16,12,18)(13,19,22,21)(17,23,20,24)(25,27,30,32)(26,29,33,38)(28,35,37,34)(31,39,40,36);(1,4,6,5)(2,3,7,8)(9,12,15,10)(11,18,14,16)(13,20,22,17)(19,24,21,23)(25,28)(26,31)(27,34)(29,36)(30,37)(32,35)(33,40)(38,39);(1,2)(3,4)(5,8)(6,7)(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,28)(26,31)(27,34)(29,36)(30,37)(32,35)(33,40)(38,39);(1,5,6,4)(2,8,7,3)(9,14,15,11)(10,18,12,16)(13,21,22,19)(17,24,20,23)(25,29)(26,32)(27,33)(28,36)(30,38)(31,35)(34,40)(37,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=ghi,d^2=h,e^2=gh,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, 8680203);