#130003

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	C₂×C₄×C₈	C₂³
prime radical	#60647	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

965 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=i,f^2=gh,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, 37199);