#130000

order	512
nilpotency class	4
isoclinism	isoclinic class of #129998
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8339
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#60648	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

889 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 889 conjugacy classes of subgroups — too many to draw. Only its 120 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37196);