#140450

order	512
nilpotency class	4
isoclinism	isoclinic class of #12463
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1675
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9615
prime radical	C₂³	#473
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

203 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 203 conjugacy classes of subgroups — too many to draw. Only its 96 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=eh,d^2=g,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dg,f^b=fi,g^b=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47646);