#142401

order	512
nilpotency class	4
isoclinism	isoclinic class of #142400
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	#135
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 357 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,g^a=gi,c^b=cf,d^b=dgi,g^b=gi,h^b=hi,d^c=dh,e^c=ei,f^c=fi,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, 49597);