#137966

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=gh,e^2=i,f^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, 45162);