#451603

order	512
nilpotency class	3
isoclinism	isoclinic class of #417571
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1113 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)(3,6,4,8)(9,10,14,16)(11,19,22,35)(12,23)(13,27,29,18)(15,28,32,21)(17,26)(20,33)(24,31)(25,34)(30,36)(37,38)(39,40);(1,3,5,4)(2,6,7,8)(9,11,14,22)(10,15,16,32)(12,24,26,39)(13,28,29,21)(17,33,23,37)(18,35,27,19)(20,30,38,25)(31,36,40,34);(1,4,5,3)(2,6,7,8)(10,16)(13,29)(17,23)(19,35)(21,28)(24,39)(25,30)(33,37);(9,12)(10,17)(11,20)(13,30)(14,26)(15,31)(16,23)(18,36)(19,37)(21,39)(22,38)(24,28)(25,29)(27,34)(32,40)(33,35)

Polycyclic presentation:

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

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, 358799);