#424588

order	512
nilpotency class	3
isoclinism	isoclinic class of #417622
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1206 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,g^c=gi,g^d=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331784);