#8922860

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

4588 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=ghi,e^2=i,f^2=h,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, 8830056);