#468370

order	512
nilpotency class	3
isoclinism	isoclinic class of #454982
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

876 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eg,c^2=g,d^2=ei,f^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,f^b=fi,d^c=de,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, 375566);