#377183

order	512
nilpotency class	3
isoclinism	isoclinic class of #24404
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	#3449	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1275 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)(3,7)(4,6)(5,8)(9,10,15,20,14,19,28,34)(11,21,25,37,24,16,30,32)(12,17,27,29,18,33,35,13)(22,31,38,39,26,36,40,23);(1,3,5,4)(2,6,8,7)(9,11,14,24)(10,16,19,21)(12,26,18,22)(13,23,29,39)(15,25,28,30)(17,31,33,36)(20,32,34,37)(27,40,35,38);(1,3,5,4)(2,7,8,6)(9,12,14,18)(10,17,19,33)(11,22)(13,20,29,34)(15,27,28,35)(16,31)(21,36)(23,37)(24,26)(25,38)(30,40)(32,39);(1,4,5,3)(2,7,8,6)(9,13,28,33,14,29,15,17)(10,18,34,27,19,12,20,35)(11,23,30,36,24,39,25,31)(16,22,37,40,21,26,32,38)

Polycyclic presentation:

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

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, 284379);