#126722

order	512
nilpotency class	4
isoclinism	isoclinic class of #11669
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#418
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#7879
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

335 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 335 conjugacy classes of subgroups — too many to draw. Only its 144 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=gi,e^2=h,f^2=g,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi,c^b=cgh,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320704, #72320710, #72320759
maximal normal subgroup of	#72320704, #72320710, #72320759

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33918);