#497510

order	512
nilpotency class	4
isoclinism	isoclinic class of #480884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×D₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

939 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,d^c=di,f^c=fi,g^c=ghi,g^f=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404706);