#146157

order	512
nilpotency class	4
isoclinism	isoclinic class of #144075
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1695
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10651
prime radical	C₂³	#473
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

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

normal subgroups only

This group has 185 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=d,c^2=dh,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,f^b=fh,h^b=hi,d^c=di,e^c=eg,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, 53353);