#129561

order	512
nilpotency class	4
isoclinism	isoclinic class of #129559
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂×#466	C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

624 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36757);