#138574

order	512
nilpotency class	4
isoclinism	isoclinic class of #138574
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9215
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 311 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=ghi,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=ehi,f^b=fhi,h^b=hi,d^c=dhi,e^c=ehi,f^c=fh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45770);