#135393

order	512
nilpotency class	4
isoclinism	isoclinic class of #135346
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1824
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

329 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 329 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=dg,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,h^a=hi,e^b=ei,d^c=di,e^c=eh,f^c=fgh,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42589);