#144918

order	512
nilpotency class	4
isoclinism	isoclinic class of #142310
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

394 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(2,3)(5,6,13,20)(7,21,27,43)(8,28,34,63)(9,19,12,16)(10,18,35,50)(11,37,36,17)(14,26,44,23)(15,39,49,31)(22,45,59,68)(24,41,38,54)(25,42,60,51)(29,64,61,40)(30,52,33,47)(32,46,55,48)(53,65,62,57)(56,67,58,66);(1,2)(3,4)(5,7,13,27)(6,14,20,44)(8,29,34,61)(9,26,12,23)(10,25,35,60)(11,38,36,24)(15,45,49,68)(16,43,19,21)(17,51,37,42)(18,41,50,54)(22,55,59,32)(28,62,63,53)(30,57,33,65)(31,58,39,56)(40,52,64,47)(46,67,48,66);(1,3,4,2)(5,8)(6,15)(7,22)(9,33)(10,32)(11,39)(12,30)(13,34)(14,40)(16,48)(17,52)(18,28)(19,46)(20,49)(21,53)(23,58)(24,61)(25,57)(26,56)(27,59)(29,38)(31,36)(35,55)(37,47)(41,45)(42,67)(43,62)(44,64)(50,63)(51,66)(54,68)(60,65)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52114);