#131398

order	512
nilpotency class	4
isoclinism	isoclinic class of #12170
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1598
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8404
prime radical	#60915	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 615 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,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,e^c=ei,f^c=fg,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38594);