#142403

order	512
nilpotency class	4
isoclinism	isoclinic class of #142402
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9921
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 353 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,g^a=gi,c^b=cf,d^b=dgi,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49599);