#135408

order	512
nilpotency class	4
isoclinism	isoclinic class of #124934
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#331
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 ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 378 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)(9,10,12,18,24,16,23,31,14,19,26,33,13,17,25,32)(11,15,20,29,35,27,34,38,22,30,37,40,21,28,36,39);(2,5)(6,8)(9,11,14,22)(10,15,19,30)(12,20,26,37)(13,21,24,35)(16,27,17,28)(18,29,33,40)(23,34,25,36)(31,38,32,39);(1,3,4,7)(2,6,5,8)(10,16)(12,23)(13,24)(15,27)(17,19)(18,33)(20,34)(21,35)(25,26)(28,30)(29,40)(36,37)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=d,e^2=h,f^2=ghi,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,h^a=hi,e^c=eh,f^c=fghi,g^c=gi

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, 42604);