#139751

order	512
nilpotency class	4
isoclinism	isoclinic class of #139751
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	#3487	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 650 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)(3,13,18,40)(4,19)(5,23,14,10)(6,11,28,37)(7,12)(8,16)(9,21,33,27)(15,43)(17,49)(20,52,41,62)(22,36)(24,31)(25,54)(26,39)(29,57)(30,48,60,55)(32,56)(34,53)(35,42)(38,50,61,45)(44,59,47,51)(46,63)(58,64);(1,3,14,16,7,18,5,24)(2,8,23,40,12,31,10,13)(4,15,41,47,22,46,20,44)(6,26,25,48,28,17,42,55)(9,29,32,61,33,58,34,38)(11,30,35,39,37,60,54,49)(19,51,62,43,36,59,52,63)(21,50,53,57,27,45,56,64);(1,4)(2,9)(3,15)(5,20)(6,27)(7,22)(8,29)(10,34)(11,36)(12,33)(13,38)(14,41)(16,47)(17,50)(18,46)(19,37)(21,28)(23,32)(24,44)(25,56)(26,45)(30,59)(31,58)(35,52)(39,63)(40,61)(42,53)(43,49)(48,64)(51,60)(54,62)(55,57)

Polycyclic presentation:

2^9|a^2=f,b^2=df,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,h^a=hi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=ei,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, 46947);