#130049

order	512
nilpotency class	4
isoclinism	isoclinic class of #130049
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8344
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#60717	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 578 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=h,f^2=gh,g^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=fghi,g^b=gi,d^c=di,e^c=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37245);