#415529

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

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

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:

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

Polycyclic presentation:

2^9|a^2=efgh,b^2=ef,c^2=e,d^2=h,e^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,f^c=fi,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, 322725);