#93565

order	512
nilpotency class	4
isoclinism	isoclinic class of #93565
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1286
commutator	C₂×C₄²	C₄²
Frattini, cosocle	C₄×#117	C₂²
socle	C₂	#7037
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₄×#117 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

294 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 62 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,e^2=f,g^2=i,b^a=bc,c^a=cf,e^a=efgh,f^a=fi,h^a=hi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 761);