#131403

order	512
nilpotency class	4
isoclinism	isoclinic class of #12304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1616
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

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

normal subgroups only

This group has 623 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281031, #72281057, #72281344
maximal normal subgroup of	#72281031, #72281057, #72281344

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38599);