#134701

order	512
nilpotency class	4
isoclinism	isoclinic class of #134701
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1809
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8600
prime radical	C₂⁴	#108
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

256 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=cgh,e^b=ei,f^b=fi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41897);