#470977

order	512
nilpotency class	3
isoclinism	isoclinic class of #458126
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

525 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)(9,10,16,20,15,21,14,22)(11,19,25,32,24,33,13,17)(12,18,27,37,29,36,28,35)(23,38,39,34,26,40,30,31);(1,3,5,4)(2,6,7,8)(9,11,14,13,15,24,16,25)(10,17,20,19,21,32,22,33)(12,23,28,30,29,26,27,39)(18,34,37,40,36,31,35,38);(2,7)(3,4)(9,12,16,27,15,29,14,28)(10,18,22,35,21,36,20,37)(11,23,25,39,24,26,13,30)(17,31,33,40,32,34,19,38);(1,4,5,3)(2,6,7,8)(9,13,14,24,15,25,16,11)(10,19,20,32,21,33,22,17)(12,26,27,30,29,23,28,39)(18,34,35,38,36,31,37,40)

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^b=di,d^c=de,e^c=ei,f^c=fi,h^c=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, 378173);