#93581

order	512
nilpotency class	4
isoclinism	isoclinic class of #93565
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1286
commutator	C₂×C₄²	C₄²
Frattini, cosocle	C₂×#405	C₂²
socle	C₂	#7039
prime radical	C₂⁴	#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#405 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

270 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 64 points:

(1,2,5,11,8,30,29,60,7,13,27,40,6,12,28,41)(3,14,16,42,17,44,52,63,19,43,51,61,18,53,25,58)(4,20,21,54,24,10,35,36,22,55,26,59,23,37,57,39)(9,31,32,47,33,50,38,49,34,56,62,15,45,46,64,48);(1,3)(2,9)(4,15,22,47)(5,25,27,52)(6,17)(7,19)(8,18)(10,14,37,43)(11,38,40,64)(12,33)(13,34)(16,29,51,28)(20,44,55,53)(21,56)(23,48,24,49)(26,31)(30,45)(32,41,62,60)(35,50)(36,63)(39,58)(42,59)(46,57)(54,61)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=fg,f^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgh,h^a=hi,c^b=cg,d^b=dh,g^b=gi,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, 777);