#146016

order	512
nilpotency class	4
isoclinism	isoclinic class of #144075
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1695
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10615
prime radical	C₂×#135	D₄
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

220 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=d,d^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,h^b=hi,e^c=eg,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53212);