#140462

order	512
nilpotency class	4
isoclinism	isoclinic class of #140462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9617
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#471
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

197 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 197 conjugacy classes of subgroups — too many to draw. Only its 92 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)(3,15)(4,22,26,53)(5,28)(6,11)(7,33)(8,14)(9,17)(10,25,40,59)(12,31)(13,30)(16,38,49,62)(18,37)(19,46)(20,42)(21,44)(23,34,50,61)(24,56,58,41)(27,60,57,39)(29,36)(32,43)(35,51)(45,48,64,47)(52,55,63,54);(1,3)(2,9)(4,23,26,50)(5,29)(6,17)(7,20)(8,21)(10,38,40,62)(11,42)(12,44)(13,36)(14,37)(15,28)(16,27,49,57)(18,32)(19,30)(22,52,53,63)(24,47,58,48)(25,54,59,55)(31,51)(33,35)(34,41,61,56)(39,45,60,64)(43,46);(1,4,6,25,7,27,31,58,8,26,32,59,30,57,5,24)(2,10,12,22,13,41,11,39,14,40,28,53,33,56,43,60)(3,16,18,48,20,23,29,55,21,49,17,47,19,50,51,54)(9,34,15,45,36,38,46,63,37,61,44,64,35,62,42,52)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=e,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ch,d^b=dg,e^b=ei,g^b=gi,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47658);