#141907

order	512
nilpotency class	4
isoclinism	isoclinic class of #13291
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1993
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9873
prime radical	C₂³	#421
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

268 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 100 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=eg,d^2=g,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49103);