#409655

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
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	#2495
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

569 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)(4,8)(5,6)(9,10,15,22,16,21,14,20)(11,23,26,36,27,37,34,17)(12,28,30,39,31,18,25,35)(13,32,24,38,29,40,33,19);(2,6)(7,8)(9,11,14,26,16,27,15,34)(10,17,22,37,21,36,20,23)(12,29,30,24,31,13,25,33)(18,38,39,40,28,19,35,32);(1,3,5,4)(2,7,6,8)(9,12,14,30,16,31,15,25)(10,18,20,35,21,28,22,39)(11,24,26,13,27,33,34,29)(17,32,23,19,36,40,37,38);(1,4,5,3)(2,8,6,7)(9,13,14,33,16,29,15,24)(10,19,22,32,21,38,20,40)(11,25,26,12,27,30,34,31)(17,35,37,18,36,39,23,28)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,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, 316851);