#125889

order	512
nilpotency class	4
isoclinism	isoclinic class of #125888
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7750
commutator, Frattini, cosocle	C₂×#97	C₂³
prime radical	#31075	C₂
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 394 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)(4,18,21,51)(5,23)(6,11)(7,13)(9,20,36,55)(10,27)(12,29)(14,42,45,62)(15,38)(16,32)(17,34)(19,52,22,39)(24,56)(25,37,54,35)(26,41)(28,33)(30,44,58,64)(31,49)(40,53)(43,50,46,60)(47,59,63,57)(48,61);(1,3,7,17)(2,8,13,34)(4,14,21,45)(5,24,26,49)(6,28,29,16)(9,30,36,58)(10,38,40,61)(11,33,12,32)(15,27,48,53)(18,50,51,60)(19,46,22,43)(20,47,55,63)(23,31,41,56)(25,44,54,64)(35,59,37,57)(39,62,52,42);(1,4)(2,9)(3,14)(5,25)(6,19)(7,21)(8,30)(10,39)(11,35)(12,37)(13,36)(15,47)(16,43)(17,45)(18,41)(20,53)(22,29)(23,51)(24,44)(26,54)(27,55)(28,46)(31,60)(32,57)(33,59)(34,58)(38,62)(40,52)(42,61)(48,63)(49,64)(50,56)

Polycyclic presentation:

2^9|a^2=f,b^2=i,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=ehi,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, 33085);