#409722

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

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

590 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)(3,7)(4,6)(5,8)(9,10)(11,17,12,26)(13,28)(14,22)(15,20)(16,21)(18,25,34,24)(19,31)(23,33,27,40)(29,35)(30,36)(32,37,39,38);(1,3,5,4)(2,6,8,7)(9,11,15,24,14,12,16,25)(10,17,20,18,22,26,21,34)(13,23,30,37,29,27,31,38)(19,32,28,33,36,39,35,40);(1,3,5,4)(2,7,8,6)(9,12,14,11)(10,18)(13,27,29,23)(15,25,16,24)(17,21)(19,33)(20,26)(22,34)(28,39)(30,38,31,37)(32,35)(36,40);(1,4,5,3)(2,6,8,7)(9,13,16,31,14,29,15,30)(10,19,20,28,22,36,21,35)(11,23,25,38,12,27,24,37)(17,32,18,33,26,39,34,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=eh,c^2=e,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 316918);