#145499

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	#33825	C₂
LCS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 215 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)(3,14)(4,8)(5,11)(6,27)(7,13)(9,17)(10,19)(12,21)(15,37)(16,46)(18,42)(20,52)(22,29)(23,31)(24,33)(25,38)(26,58)(28,53)(30,51)(32,55)(34,49)(35,57)(36,48)(39,54)(40,43)(41,45)(44,59)(47,60)(50,61)(56,62)(63,64);(2,8)(3,15)(4,19)(6,21)(9,32)(10,12)(11,29)(13,31)(14,40)(16,20)(17,44)(18,45)(22,48)(23,51)(24,49)(26,54)(27,30)(28,33)(34,35)(36,39)(37,46)(38,62)(41,52)(42,61)(47,53)(56,64)(58,63)(59,60);(1,3,4,20,21,45,51,43,7,18,23,16,6,15,19,50)(2,9,10,35,27,59,31,60,13,34,30,32,12,33,8,28)(5,24,22,55,54,49,64,47,25,44,56,57,26,17,48,53)(11,37,36,46,58,42,62,40,38,41,63,52,39,14,29,61)

Polycyclic presentation:

2^9|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, 52695);