#125212

order	512
nilpotency class	4
isoclinism	isoclinic class of #125143
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#333
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

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

normal subgroups only

This group has 347 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)(9,10,13,19,14,18,30,37,15,21,12,17,27,20,28,36)(11,22,23,38,25,16,26,32,24,33,39,35,31,34,29,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,24)(10,16)(12,26,13,29)(14,31,27,25)(17,35)(18,22)(19,38)(20,33)(21,34)(23,30,39,28)(32,36)(37,40);(9,12,14,28,15,13,27,30)(10,17,18,36,21,19,20,37)(11,23,25,26,24,39,31,29)(16,32,33,35,34,40,22,38)

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=h,d^2=gi,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=efg,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, 32408);