#152569

order	512
nilpotency class	5
isoclinism	isoclinic class of #152569
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13041
commutator	#124	C₂²×C₄
Frattini, cosocle	C₂×#124	C₂³
prime radical	#33568	C₂
LCS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#124 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

523 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 60 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)(3,8,14,32)(4,17,18,53)(5,22,25,40)(6,27,28,12)(7,10)(9,19,35,54)(11,26)(13,41,42,64)(15,47,45,38)(16,33)(20,52,21,39)(23,57)(24,34,56,36)(29,37)(30,48,61,55)(31,50)(43,60,58,62)(44,63,49,51)(46,59);(1,3)(2,8,10,32)(4,13)(5,23)(6,16)(7,14)(9,30,36,62)(11,38,37,47)(12,33,27,59)(15,29)(17,51,39,64)(18,49)(19,55)(20,42)(21,44)(22,31,40,57)(24,58)(25,50)(26,45)(28,46)(34,61,35,60)(41,52,63,53)(43,54)(48,56);(1,4)(2,9)(3,13)(5,24)(6,18)(7,21)(8,30)(10,36)(11,39)(12,35)(14,44)(15,48)(16,49)(17,37)(19,25)(20,28)(22,52)(23,58)(26,54)(27,34)(29,56)(31,63)(32,62)(33,60)(38,64)(40,53)(41,57)(42,46)(43,45)(47,51)(50,55)(59,61)

Polycyclic presentation:

2^9|a^2=f,b^2=df,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,d^b=di,e^b=eghi,f^b=fi,h^b=hi,d^c=dg,e^c=ei,f^c=fg,g^c=gi,h^c=hi,e^d=ehi,g^d=gi,f^e=fhi,g^e=gi,g^f=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59765);