#138087

order	512
nilpotency class	4
isoclinism	isoclinic class of #138086
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9119
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2875 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

363 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,d^2=ghi,e^2=i,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,h^a=hi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,d^c=di,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45283);