#455205

order	512
nilpotency class	3
isoclinism	isoclinic class of #455110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=i,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,g^c=gi,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, 362401);