#130005

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	#60649	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

913 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)(9,10,13,18,15,20,14,19)(11,21,24,34,25,38,23,37)(12,26,28,17,30,35,29,36)(16,27,32,22,33,39,31,40);(2,5)(3,7)(9,11)(10,16)(12,27)(13,23)(14,24)(15,25)(17,34)(18,31)(19,32)(20,33)(21,35)(22,29)(26,38)(28,40)(30,39)(36,37);(9,10)(11,16)(12,17)(13,19)(14,18)(15,20)(21,22)(23,32)(24,31)(25,33)(26,28)(27,34)(29,35)(30,36)(37,39)(38,40)

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=i,f^2=ghi,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, 37201);