#409634

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

591 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:

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

Polycyclic presentation:

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

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, 316830);