#152360

order	512
nilpotency class	5
isoclinism	isoclinic class of #149797
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#127	C₂³
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#535 ◁ G
LECS	G ▷ C₂×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

503 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 503 conjugacy classes of subgroups — too many to draw. Only its 62 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)(11,15)(12,23,25,16)(13,18,28,17)(14,19)(20,35,36,40)(21,31)(22,30,37,34)(24,39,38,29)(26,32,27,33);(2,5)(6,8)(9,11)(10,15)(12,24)(13,22)(14,21)(16,29)(17,34)(18,30)(19,31)(20,26)(23,39)(25,38)(27,36)(28,37)(32,35)(33,40);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,20)(13,27)(14,26)(15,29)(17,32)(18,23)(19,33)(21,24)(22,38)(25,28)(30,39)(31,40)(34,35)(36,37)

Polycyclic presentation:

2^9|a^2=f,c^2=d,e^2=hi,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ch,e^b=ei,e^c=ehi,f^c=fgh,g^c=gi,f^e=fhi,g^f=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59556);