#411232

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

740 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)(9,10,15,19,14,20,16,30)(11,17,31,35,23,36,24,34)(12,25,27,18,28,39,13,29)(21,37,26,40,22,33,38,32);(9,11,16,31,14,23,15,24)(10,17,30,35,20,36,19,34)(12,26,27,21,28,38,13,22)(18,33,39,40,29,37,25,32);(2,5)(6,8)(9,12,16,27,14,28,15,13)(10,18,19,25,20,29,30,39)(11,21,24,26,23,22,31,38)(17,32,35,33,36,40,34,37);(1,3,4,7)(2,6,5,8)(9,13,15,28,14,27,16,12)(10,18,19,25,20,29,30,39)(11,22,24,38,23,21,31,26)(17,33,34,32,36,37,35,40)

Polycyclic presentation:

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

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, 318428);