#93780

order	512
nilpotency class	5
isoclinism	isoclinic class of #93780
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7076
commutator	C₂×M₃₂	C₂×C₄
Frattini, cosocle	#2112	C₂²
prime radical	#13309	C₂
LCS	G ▷ C₂×M₃₂ ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2112 ◁ G
LECS	G ▷ #2112 ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

200 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 28 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 — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,c^2=fh,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=defh,e^b=eh,f^b=fh,h^b=hi,d^c=dghi,e^c=ei

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, 976);