#92883

order	512
nilpotency class	4
isoclinism	isoclinic class of #7076
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#327
commutator	C₂²×C₈	C₂×C₈
Frattini, cosocle	C₂×#322	C₂²
socle	C₂²	#1271
prime radical	C₂³	D₄⋊C₈
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×#322 ▷ C₂×C₄² ▷ C₂² ▷ C₁

150 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 128

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=g,f^2=i,g^2=hi,b^a=bc,c^a=ce,e^a=eh,c^b=cf,d^b=defh,f^b=fi,g^b=gi,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 79);