#454771

order	512
nilpotency class	3
isoclinism	isoclinic class of #454039
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1251 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)(6,8)(9,10,14,20,13,19,15,31)(11,21,25,35,24,16,26,34)(12,27,29,18,28,37,30,17)(22,36,40,32,39,33,23,38);(1,2,4,5)(3,6,7,8)(9,11,13,24)(10,16,19,21)(12,23,28,40)(14,25,15,26)(17,36,18,33)(20,34,31,35)(22,30,39,29)(27,32,37,38);(1,3)(2,6)(4,7)(5,8)(9,12,13,28)(10,17,19,18)(11,22,24,39)(14,30,15,29)(16,32,21,38)(20,37,31,27)(23,26,40,25)(33,34,36,35);(3,7)(6,8)(9,12)(10,18)(11,23)(13,28)(14,29)(15,30)(16,33)(17,19)(20,37)(21,36)(22,25)(24,40)(26,39)(27,31)(32,35)(34,38)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361967);