#413326

order	512
nilpotency class	3
isoclinism	isoclinic class of #398436
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1290 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)(11,17,24,33)(12,14,19,16)(13,26,28,25)(18,27,34,29)(21,30,32,23)(22,36,38,35)(31,39,40,37);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,21)(13,22)(14,30)(15,24)(16,23)(18,31)(19,32)(20,33)(25,35)(26,36)(27,39)(28,38)(29,37)(34,40);(1,4,5,3)(2,6,7,8)(9,12,14,20,15,19,16,10)(11,21,23,17,24,32,30,33)(13,25,27,34,28,26,29,18)(22,35,37,31,38,36,39,40);(9,13,14,27,15,28,16,29)(10,18,12,25,20,34,19,26)(11,22,30,39,24,38,23,37)(17,31,21,35,33,40,32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320522);