#131355

order	512
nilpotency class	4
isoclinism	isoclinic class of #131354
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

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

normal subgroups only

This group has 723 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=hi,e^2=i,f^2=h,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,e^c=ei,f^c=fgi,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, 38551);