#152944

order	512
nilpotency class	5
isoclinism	isoclinic class of #152939
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13223
commutator	C₂×C₄²	C₂²×C₄
Frattini, prime radical, cosocle	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

379 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 379 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=ei,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dgh,f^b=fgh,g^b=ghi,h^b=hi,f^d=fhi,g^f=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60140);