#125643

order	512
nilpotency class	4
isoclinism	isoclinic class of #125642
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7734
commutator, Frattini, cosocle	C₂×#97	C₂³
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 461 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)(3,14)(4,10)(5,24)(6,30)(7,12)(8,13)(9,16)(11,28)(15,45)(17,37)(18,43)(19,44)(20,38)(21,41)(22,35)(23,36)(25,40)(26,58)(27,56)(29,39)(31,46)(32,51)(33,52)(34,55)(42,47)(48,64)(49,63)(50,59)(53,61)(54,62)(57,60);(1,3)(2,9)(4,15)(5,25)(6,17)(7,18)(8,19)(10,31)(11,37)(12,40)(13,33)(14,24)(16,28)(20,53)(21,47)(22,48)(23,49)(26,57)(27,51)(29,52)(30,32)(34,42)(35,61)(36,60)(38,45)(39,43)(41,59)(44,56)(46,55)(50,54)(58,64)(62,63);(1,4,8,23)(2,10,13,36)(3,15,19,49)(5,26,27,55)(6,22,7,21)(9,31,33,60)(11,38,39,62)(12,41,30,35)(14,42,44,64)(16,50,52,53)(17,48,18,47)(20,28,54,29)(24,34,56,58)(25,57,51,46)(32,61,40,59)(37,45,43,63)

Polycyclic presentation:

2^9|c^2=i,d^2=fi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,d^b=df,e^b=egi,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=eh,f^c=fi,g^c=gi,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, 32839);