#125144

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

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	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

274 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=fgi,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, 32340);