#139758

order	512
nilpotency class	4
isoclinism	isoclinic class of #139755
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9495
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#3485	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

659 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 659 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dgi,e^b=eh,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,f^c=fh,g^c=gi,f^d=fi,f^e=fi

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, 46954);