#470976

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	#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

535 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=fh,c^2=ef,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,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, 378172);