#136421

order	512
nilpotency class	4
isoclinism	isoclinic class of #136420
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8894
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

323 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgh,d^2=gi,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=dg,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=eh,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43617);