#468885

order	512
nilpotency class	3
isoclinism	isoclinic class of #454982
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1034 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:

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

Polycyclic presentation:

2^9|a^2=fg,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, 376081);