#152260

order	512
nilpotency class	5
isoclinism	isoclinic class of #149799
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2128
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#532	C₂³
socle	C₂	#12799
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2932 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

317 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,14)(4,19,22,9)(5,24)(6,12,28,11)(7,13)(8,16)(10,26)(15,41,44,62)(17,40)(18,39,47,61)(20,51,48,38)(21,53,54,55)(23,33,27,34)(25,35,57,36)(29,52,49,64)(30,60,37,46)(31,45)(32,42,50,56)(43,58,63,59);(1,3,5,16,7,17,26,45)(2,8,10,14,13,31,24,40)(4,20,21,52,23,15,42,43)(6,18,25,46,28,47,57,60)(9,29,32,38,34,59,55,62)(11,37,35,61,12,30,36,39)(19,49,50,51,33,58,53,41)(22,48,54,64,27,44,56,63);(1,4)(2,9)(3,15)(5,21)(6,27)(7,23)(8,29)(10,32)(11,33)(12,19)(13,34)(14,38)(16,43)(17,20)(18,48)(22,28)(24,55)(25,56)(26,42)(30,49)(31,59)(35,53)(36,50)(37,58)(39,51)(40,62)(41,61)(44,47)(45,52)(46,64)(54,57)(60,63)

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=i,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,c^b=ch,e^b=ei,e^c=ehi,f^c=fgh,g^c=gi,f^e=fhi,g^f=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59456);