#128070

order	512
nilpotency class	4
isoclinism	isoclinic class of #128069
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8101
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

1279 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1279 conjugacy classes of subgroups — too many to draw. Only its 140 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)(3,13,16,38)(4,9,19,30)(5,10,22,34)(7,12,24,36)(8,25,27,15)(14,39,41,63)(17,32,45,58)(18,29,46,56)(20,31,47,57)(21,33,48,59)(23,35,49,60)(26,52,53,40)(28,51,55,44)(37,61,62,42)(43,54,50,64);(1,3)(2,8)(4,14)(5,15)(6,16,24,45)(7,17)(9,26)(10,32)(11,27,36,55)(12,28)(13,35)(18,40)(19,41,47,61)(20,42)(21,43)(22,25,49,51)(23,44)(29,37)(30,53,57,64)(31,54)(33,39)(34,58,60,38)(46,52,48,50)(56,62,59,63);(1,4,7,20)(2,9,12,31)(3,14,17,42)(5,21,23,18)(6,19,24,47)(8,26,28,54)(10,33,35,29)(11,30,36,57)(13,37,32,39)(15,43,44,40)(16,41,45,61)(22,48,49,46)(25,50,51,52)(27,53,55,64)(34,59,60,56)(38,62,58,63)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=i,b^a=bd,c^a=cg,d^a=dg,f^a=fhi,h^a=hi,d^b=dh,e^b=egi,g^b=gi,d^c=di,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313150, #72313154, #72313161
maximal normal subgroup of	#72313150, #72313154, #72313161

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35266);