#138949

order	512
nilpotency class	4
isoclinism	isoclinic class of #138944
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9259
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#131	C₂³
prime radical	#33524	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

506 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=i,d^2=gh,e^2=gh,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,d^c=dgh,e^c=eghi,f^c=fgh,h^c=hi,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46145);