#138572

order	512
nilpotency class	4
isoclinism	isoclinic class of #12465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1677
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9215
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

305 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=ghi,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ehi,f^b=fhi,h^b=hi,d^c=dhi,e^c=eh,f^c=fh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45768);