#145497

order	512
nilpotency class	4
isoclinism	isoclinic class of #145496
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10512
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

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

normal subgroups only

This group has 220 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=d,d^2=gi,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,g^a=gi,c^b=cf,d^b=dg,f^b=fh,g^b=gi,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, 52693);