#152249

order	512
nilpotency class	5
isoclinism	isoclinic class of #149792
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2126
commutator	C₂×C₄²	C₂²×C₄
Frattini, prime radical, cosocle	#532	C₂³
socle	C₂	#12798
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 293 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=h,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,e^c=eh,f^c=fgh,g^c=gi,f^e=fhi,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, 59445);