#418643

order	512
nilpotency class	3
isoclinism	isoclinic class of #417487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1436 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 325839);