#385293

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

884 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)(3,4)(5,7)(6,8)(9,10,16,21,14,20,15,32)(11,18,12,25,23,17,24,34)(13,27,31,19,29,37,30,36)(22,38,26,35,28,40,39,33);(1,3)(2,6)(4,7)(5,8)(9,11)(10,17)(12,16)(13,28)(14,23)(15,24)(18,20)(19,35)(21,34)(22,29)(25,32)(26,30)(27,38)(31,39)(33,36)(37,40);(1,4,5,6)(2,3,7,8)(9,12,16,23,14,24,15,11)(10,18,32,34,20,17,21,25)(13,26,31,28,29,39,30,22)(19,35,27,38,36,33,37,40);(2,7)(3,8)(9,13,15,30,14,29,16,31)(10,19,21,37,20,36,32,27)(11,22,24,39,23,28,12,26)(17,33,34,38,18,35,25,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 292489);