#153544

order	512
nilpotency class	6
isoclinism	isoclinic class of #153539
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13584
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	#494	C₂³
prime radical	#532	D₄
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #3152 ◁ G
LECS	G ▷ #494 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

257 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 257 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=gh,e^2=i,f^2=hi,g^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,c^b=chi,d^b=dghi,e^b=efg,f^b=fhi,g^b=gi,d^c=di,e^d=egh,f^e=fhi,g^e=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60740);