#143696

order	512
nilpotency class	4
isoclinism	isoclinic class of #143695
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10228
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#28249	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

430 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=gi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,f^b=fi,g^b=gi,d^c=di,e^c=eh,f^c=fi,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, 50892);