#125174

order	512
nilpotency class	4
isoclinism	isoclinic class of #125120
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 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

413 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 413 conjugacy classes of subgroups — too many to draw. Only its 94 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)(3,7,8,5)(4,6)(9,10,13,19,14,18,28,36,15,21,29,35,27,20,12,17)(11,22,24,37,25,34,38,40,26,33,39,31,30,16,23,32);(1,3)(2,5)(4,8)(6,7)(9,11,15,26)(10,16,21,34)(12,24,28,39)(13,23,29,38)(14,30,27,25)(17,32,36,40)(18,33,20,22)(19,31,35,37);(9,12,27,29,15,28,14,13)(10,17,20,35,21,36,18,19)(11,23,30,39,26,38,25,24)(16,31,33,40,34,37,22,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32370);