#8922550

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

4541 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8829746);