#146044

order	512
nilpotency class	4
isoclinism	isoclinic class of #146044
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10619
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
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

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

normal subgroups only

This group has 222 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:

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

Polycyclic presentation:

2^9|a^2=eg,c^2=d,e^2=g,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,e^b=ei,f^b=fh,h^b=hi,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, 53240);