#137399

order	512
nilpotency class	4
isoclinism	isoclinic class of #12173
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	#9041
prime radical	#33530	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

564 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=g,e^2=g,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=egh,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=egi,f^c=fh,g^c=gi,h^c=hi,e^d=ei,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281031, #72281125, #72281216
maximal normal subgroup of	#72281031, #72281125, #72281216

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44595);