#512683

order	512
nilpotency class	5
isoclinism	isoclinic class of #512669
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄⋊Q₈	C₂⁴
prime radical	#3500	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #3500 ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

791 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 791 conjugacy classes of subgroups — too many to draw. Only its 184 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=ehi,f^b=fgh,g^b=gi,d^c=di,e^c=ei,f^e=fhi,g^f=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419879);