#126350

order	512
nilpotency class	4
isoclinism	isoclinic class of #126349
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7791
commutator, Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	#455
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 247 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=i,d^2=f,e^2=hi,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=ehi,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, 33546);