#474191

order	512
nilpotency class	3
isoclinism	isoclinic class of #458125
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	#2571
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

556 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:

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

Polycyclic presentation:

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

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, 381387);