#143964

order	512
nilpotency class	4
isoclinism	isoclinic class of #143963
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10260
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33594	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

414 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 80 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)(3,18)(4,24)(5,32)(6,36)(7,14)(8,16)(9,15)(10,17)(11,20)(12,27)(13,35)(19,53)(21,42)(22,59)(23,40)(25,52)(26,46)(28,45)(29,50)(30,44)(31,49)(33,51)(34,47)(37,41)(38,48)(39,56)(43,58)(54,63)(55,61)(57,62)(60,64);(1,3)(2,11)(4,25)(5,33)(6,21)(7,37)(8,38)(9,22)(10,23)(12,43)(13,48)(14,40)(15,51)(16,41)(17,42)(18,32)(19,28)(20,35)(24,61)(26,56)(27,60)(29,57)(30,58)(31,54)(34,55)(36,59)(39,44)(45,63)(46,52)(47,62)(49,64)(50,53);(1,4,10,31)(2,12,17,47)(3,19,23,56)(5,34,7,27)(6,28,35,26)(8,30,9,29)(11,39,42,53)(13,49,36,24)(14,50,32,44)(15,46,16,45)(18,52,40,63)(20,57,21,58)(22,60,38,55)(25,33,54,37)(41,64,51,61)(43,59,62,48)

Polycyclic presentation:

2^9|c^2=i,d^2=gi,e^2=hi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,f^b=fg,g^b=gi,d^c=dg,e^c=eh,f^c=fg,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, 51160);