#153065

order	512
nilpotency class	5
isoclinism	isoclinic class of #149923
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#354
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#124	C₂³
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#124 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

318 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 66 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)(4,5)(6,8)(9,10,13,18,12,16,20,32)(11,15,23,30,22,29,38,40)(14,17,27,28,24,19,26,35)(21,31,36,33,37,34,25,39);(2,5)(6,7)(9,11,20,36)(10,15,28,40)(12,22,13,25)(14,21,26,38)(16,29,35,30)(17,31,32,39)(18,33,19,34)(23,24,37,27);(1,3,4,8)(2,6,5,7)(10,16)(15,29)(17,19)(18,32)(28,35)(30,40)(31,34)(33,39)

Polycyclic presentation:

2^9|a^2=f,b^2=defh,c^2=ei,d^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dgh,f^b=fghi,g^b=ghi,h^b=hi,f^d=fhi,g^f=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60261);