#138264

order	512
nilpotency class	4
isoclinism	isoclinic class of #138264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9147
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

699 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 699 conjugacy classes of subgroups — too many to draw. Only its 100 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=gh,e^2=i,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,d^c=di,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45460);