#500454

order	512
nilpotency class	4
isoclinism	isoclinic class of #499754
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61251	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

796 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,g^c=gh,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, 407650);