#145942

order	512
nilpotency class	4
isoclinism	isoclinic class of #144056
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1697
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10593
prime radical	C₂³	#471
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

201 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 201 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53138);