#129791

order	512
nilpotency class	4
isoclinism	isoclinic class of #129789
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#60458	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

905 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,f^b=fghi,g^b=gi,h^b=hi,e^c=ei,f^c=fi

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, 36987);