#382564

order	512
nilpotency class	3
isoclinism	isoclinic class of #29751
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×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1858 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)(3,6)(4,5)(7,8)(9,10,13,19,15,18,14,20)(11,16,22,32,23,31,29,36)(12,24,26,34,27,35,28,17)(21,37,38,40,39,33,25,30);(2,5)(6,8)(9,11)(10,16)(12,25)(13,22)(14,29)(15,23)(17,33)(18,31)(19,32)(20,36)(21,26)(24,30)(27,38)(28,39)(34,37)(35,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,21,23,39)(13,28)(14,26)(15,27)(16,30,31,40)(18,34)(19,35)(20,24)(22,25,29,38)(32,33,36,37);(3,7)(6,8)(10,18)(16,31)(17,34)(19,20)(24,35)(30,40)(32,36)(33,37)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289760);