#93792

order	512
nilpotency class	5
isoclinism	isoclinic class of #93792
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7077
commutator	C₂×M₃₂	C₂×C₄
Frattini, cosocle	#2112	C₂²
prime radical	C₂⁴	SD₃₂
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

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

normal subgroups only

This group has 131 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=g,c^2=fhi,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=defhi,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, 988);