#137965

order	512
nilpotency class	4
isoclinism	isoclinic class of #137964
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9113
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

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

normal subgroups only

This group has 557 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)(3,15,20,37)(4,22,25,41)(5,26,29,39)(6,12)(7,13)(8,14)(9,18)(10,24,36,30)(11,16,40,28)(17,47)(19,43,21,44)(23,38)(27,35)(31,59,62,64)(32,45)(33,52)(34,53)(42,54)(46,56)(48,55,51,61)(49,60)(50,63)(57,58);(1,3,16,45,6,21,5,18,8,20,28,52,7,19,29,53)(2,9,26,44,13,32,40,15,14,34,39,43,12,33,11,37)(4,17,46,64,30,49,27,51,25,50,57,59,24,54,23,48)(10,31,58,47,22,55,35,42,36,62,56,63,41,61,38,60);(1,4)(2,10)(3,17)(5,27)(6,30)(7,24)(8,25)(9,31)(11,38)(12,41)(13,22)(14,36)(15,42)(16,46)(18,51)(19,54)(20,50)(21,49)(23,29)(26,58)(28,57)(32,55)(33,61)(34,62)(35,40)(37,60)(39,56)(43,63)(44,47)(45,64)(48,53)(52,59)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=gh,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,h^a=hi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,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, 45161);