#330582

order	512
nilpotency class	3
isoclinism	isoclinic class of #329587
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

517 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=efh,b^2=i,c^2=i,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gi,d^c=de,f^c=fh,g^c=gi

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, 237778);