#471375

order	512
nilpotency class	3
isoclinism	isoclinic class of #461251
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

542 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378571);