#483022

order	512
nilpotency class	4
isoclinism	isoclinic class of #483021
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1533
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#97	C₂⁴
prime radical	#3499	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2875 ◁ G
LECS	G ▷ #97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

686 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 192 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=h,f^2=g,g^2=i,b^a=be,c^a=ch,e^a=eg,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,d^c=di,f^c=fi,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390218);