#136216

order	512
nilpotency class	4
isoclinism	isoclinic class of #12307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1616
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8870
prime radical	#3054	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 362 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)(3,16,20,44)(4,22,25,42)(5,27)(6,11)(7,14)(8,15)(9,18,35,52)(10,24,39,58)(12,28)(13,32)(17,47)(19,36,53,37)(21,46,30,34)(23,41,57,40)(26,38,31,56)(29,43)(33,50)(45,63)(48,61)(49,62)(51,59)(54,60)(55,64);(1,3,7,30,8,20,32,21)(2,9,13,37,15,35,14,36)(4,17,31,50,25,51,26,54)(5,19,6,18,28,53,29,52)(10,33,41,47,39,60,40,59)(11,34,27,16,43,46,12,44)(22,55,38,62,42,61,56,45)(23,49,58,64,57,63,24,48);(1,4)(2,10)(3,17)(5,24)(6,23)(7,31)(8,25)(9,33)(11,42)(12,38)(13,41)(14,40)(15,39)(16,45)(18,49)(19,48)(20,51)(21,54)(22,43)(26,32)(27,56)(28,58)(29,57)(30,50)(34,61)(35,60)(36,59)(37,47)(44,62)(46,55)(52,63)(53,64)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,d^2=g,e^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dg,e^b=ehi,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=egi,f^c=fi,g^c=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280998, #72281031, #72281216
maximal normal subgroup of	#72280998, #72281031, #72281216

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43412);