#131353

order	512
nilpotency class	4
isoclinism	isoclinic class of #131350
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8403
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	#60773	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

719 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 719 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=eg,f^b=fgh,h^b=hi,d^c=dgi,f^c=fg,e^d=ei,f^d=fi,f^e=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, 38549);