#369400

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1460 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276596);