#126701

order	512
nilpotency class	4
isoclinism	isoclinic class of #126701
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1735
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#7875
prime radical	C₂⁴	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#97 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

353 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 152 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=gi,e^2=h,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fg,g^a=gi,d^b=di,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33897);