#125946

order	512
nilpotency class	4
isoclinism	isoclinic class of #125945
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7754
commutator, Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	#453
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

290 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 54 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)(3,8)(4,18,21,52)(5,23)(6,11)(7,13)(9,20,36,56)(10,27)(12,29)(14,42,45,64)(15,38)(16,32)(17,34)(19,39,22,51)(24,57)(25,35,53,37)(26,41)(28,33)(30,44,60,58)(31,49)(40,55)(43,50,46,63)(47,59,54,61)(48,62);(1,3,7,17)(2,8,13,34)(4,14,21,45)(5,24,26,49)(6,28,29,16)(9,30,36,60)(10,38,40,62)(11,33,12,32)(15,27,48,55)(18,50,52,63)(19,46,22,43)(20,54,56,47)(23,31,41,57)(25,58,53,44)(35,61,37,59)(39,42,51,64);(1,4,7,21)(2,9)(3,14,17,45)(5,25,26,53)(6,19,29,22)(8,30)(10,39)(11,35)(12,37)(13,36)(15,47,48,54)(16,43,28,46)(18,23)(20,55,56,27)(24,58,49,44)(31,50)(32,59)(33,61)(34,60)(38,42)(40,51)(41,52)(57,63)(62,64)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=f,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,h^a=hi,d^b=df,e^b=egi,g^b=gi,h^b=hi,d^c=dg,e^c=eh,f^c=fi,g^c=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33142);