#431975

order	512
nilpotency class	3
isoclinism	isoclinic class of #418484
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23426	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1188 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 339171);