#135388

order	512
nilpotency class	4
isoclinism	isoclinic class of #124681
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1514
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

348 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 84 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)(11,21)(12,26)(13,28,30,17)(14,18)(15,20)(16,22)(19,31,35,27)(23,36,29,33)(24,34)(25,37)(32,39,40,38);(2,5)(6,8)(9,11,15,25)(10,16)(12,22,14,24)(13,29,30,23)(17,32)(18,21)(19,36)(20,34)(26,37)(27,39,31,38)(28,40)(33,35);(1,3,4,7)(2,6,5,8)(10,17,20,28)(12,14)(13,27)(16,32,34,40)(18,35,26,19)(21,33,37,36)(22,24)(23,38)(29,39)(30,31)

Polycyclic presentation:

2^9|a^2=f,b^2=ghi,c^2=dg,d^2=i,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,h^a=hi,e^b=ei,d^c=di,e^c=eh,f^c=fgh,g^c=gi,h^c=hi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307234, #72307318, #72307336
maximal normal subgroup of	#72307234, #72307318, #72307336

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42584);