#135409

order	512
nilpotency class	4
isoclinism	isoclinic class of #124861
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1558
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

370 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 64

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

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

Polycyclic presentation:

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

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, 42605);