#124504

order	512
nilpotency class	4
isoclinism	isoclinic class of #11769
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1505
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×#97	C₂³
socle	C₂	#7591
prime radical	C₂³	#421
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 261 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fg,c^2=fgh,d^2=g,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=cf,d^a=df,g^a=gi,d^b=dg,e^b=efg,f^b=fi,g^b=gi,d^c=di,e^c=ei,e^d=ei

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, 31700);