#153454

order	512
nilpotency class	6
isoclinism	isoclinic class of #153448
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1402
commutator	C₄⋊Q₈	C₂²×C₄
Frattini, cosocle	#494	C₂³
prime radical	#3544	C₄
LCS	G ▷ C₄⋊Q₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#494 ◁ G
LECS	G ▷ #494 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

305 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 48 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:

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

Polycyclic presentation:

2^9|a^2=e,d^2=i,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=ci,d^b=di,e^b=efh,f^b=fhi,g^b=ghi,e^d=egh,f^d=fhi,g^d=gi,f^e=fhi,g^e=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, 60650);