#443498

order	512
nilpotency class	3
isoclinism	isoclinic class of #420420
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1253 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)(6,7)(9,10)(11,22)(12,26)(13,30)(14,20)(15,33)(16,21)(17,24)(18,23)(19,27)(25,36)(28,35)(29,39)(31,38)(32,40)(34,37);(1,2,4,5)(3,7,8,6)(9,11,14,24)(10,17,20,22)(12,27,28,32)(13,29,31,23)(15,25,16,34)(18,38,39,30)(19,26,40,35)(21,36,33,37);(9,12,16,23,14,28,15,29)(10,18,21,35,20,39,33,26)(11,13,25,27,24,31,34,32)(17,19,37,38,22,40,36,30);(1,3,4,8)(2,6,5,7)(9,13,14,31)(10,19,20,40)(11,23,24,29)(12,25,28,34)(15,32,16,27)(17,35,22,26)(18,37,39,36)(21,38,33,30)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 350694);