#145152

order	512
nilpotency class	4
isoclinism	isoclinic class of #145152
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10431
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#28245	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 329 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)(3,10)(4,20)(5,27)(6,12)(7,32)(8,13)(9,14)(11,23)(15,48)(16,51)(17,36)(18,38)(19,39)(21,34)(22,42)(24,43)(25,45)(26,57)(28,46)(29,60)(30,44)(31,41)(33,37)(35,53)(40,55)(47,59)(49,64)(50,63)(52,61)(54,62)(56,58);(1,3)(2,10)(4,21)(5,17)(6,30)(7,18)(8,33)(9,19)(11,40)(12,44)(13,37)(14,39)(15,24)(16,29)(20,56)(22,58)(23,50)(25,54)(26,52)(27,36)(28,53)(31,55)(32,38)(34,42)(35,46)(41,63)(43,64)(45,61)(47,48)(49,59)(51,60)(57,62);(1,4,7,26,9,22,8,25)(2,11,13,47,14,41,32,43)(3,15,18,55,19,49,33,50)(5,24,6,31,29,59,28,23)(10,34,37,62,39,56,38,61)(12,45,27,42,46,57,60,20)(16,52,53,58,17,54,30,21)(35,63,51,64,44,40,36,48)

Polycyclic presentation:

2^9|c^2=dg,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=cf,e^b=ei,f^b=fh,h^b=hi,d^c=di,e^c=egi,f^c=fh,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, 52348);