#145500

order	512
nilpotency class	4
isoclinism	isoclinic class of #145499
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10513
commutator, Frattini, cosocle	#405	C₂³
prime radical	#3176	C₂²
LCS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

209 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 209 conjugacy classes of subgroups — too many to draw. Only its 48 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=dh,d^2=gi,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=cf,d^b=dg,f^b=fh,g^b=gi,d^c=di,e^c=egh,f^c=fgh,h^c=hi,f^e=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52696);