#497531

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	#38095	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#548 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

983 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)(9,10,14,20,15,21,12,18)(11,16,25,31,22,32,23,30)(13,19,26,35,27,17,28,34)(24,29,37,39,36,33,38,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,23)(13,24)(14,25)(15,22)(17,33)(18,30)(19,29)(20,31)(21,32)(26,37)(27,36)(28,38)(34,40)(35,39);(1,3,4,7)(2,6,5,8)(10,17)(11,22)(13,27)(16,29)(18,34)(19,21)(20,35)(23,25)(26,28)(30,39)(31,40)(32,33);(2,5)(6,8)(9,12,15,14)(10,18,21,20)(11,23,22,25)(13,26,27,28)(16,30,32,31)(17,34,19,35)(24,37,36,38)(29,39,33,40)

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,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, 404727);