#145496

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

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

normal subgroups only

This group has 226 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)(6,26)(7,12)(9,17)(10,19)(13,20)(15,36)(16,45)(18,41)(21,28)(22,29)(23,33)(24,38)(25,54)(27,47)(30,50)(31,59)(32,43)(34,49)(35,48)(37,51)(39,42)(40,44)(46,60)(52,55)(53,58)(56,63)(57,64)(61,62);(2,8)(3,15)(4,19)(6,20)(9,31)(10,13)(11,28)(12,29)(14,39)(16,46)(17,43)(18,44)(21,48)(22,50)(23,49)(25,51)(26,30)(27,32)(33,56)(34,55)(35,37)(36,45)(38,58)(40,60)(41,61)(47,63)(53,64)(54,57);(1,3,4,16,20,44,50,62,7,18,22,46,6,15,19,42)(2,9,10,31,26,32,29,56,12,34,30,55,13,33,8,27)(5,23,21,52,51,49,64,63,24,43,53,59,25,17,48,47)(11,36,35,60,54,41,58,61,38,40,57,45,37,14,28,39)

Polycyclic presentation:

2^9|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, 52692);