#512637

order	512
nilpotency class	5
isoclinism	isoclinic class of #512636
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#354
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#127	C₂⁴
prime radical	#3499	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×Q₈ ◁ C₂×#41 ◁ #3499 ◁ G
LECS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 790 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)(3,14,18,30)(4,9)(5,10)(6,12,26,11)(7,13)(8,17,29,25)(15,39,43,55)(16,40,46,58)(19,31)(20,33,49,32)(21,34)(22,35)(23,37,52,36)(24,38)(27,42,54,48)(28,45,57,51)(41,61,63,56)(44,62,64,53)(47,60,50,59);(1,3,7,17)(2,8)(4,15,21,42)(5,16,24,45)(6,25,26,18)(9,27)(10,28)(11,14)(12,29)(13,30)(19,41,22,44)(20,48,49,43)(23,51,52,46)(31,53)(32,39)(33,54)(34,55)(35,56)(36,40)(37,57)(38,58)(47,64,50,63)(59,61)(60,62);(1,4,7,21)(2,9,13,34)(3,15,17,42)(5,22,24,19)(6,20,26,49)(8,27,30,55)(10,35,38,31)(11,32,12,33)(14,39,29,54)(16,44,45,41)(18,43,25,48)(23,50,52,47)(28,56,58,53)(36,60,37,59)(40,62,57,61)(46,64,51,63);(1,5,7,24)(2,10,13,38)(3,16,17,45)(4,19,21,22)(6,23,26,52)(8,28,30,58)(9,31,34,35)(11,36,12,37)(14,40,29,57)(15,41,42,44)(18,46,25,51)(20,47,49,50)(27,53,55,56)(32,59,33,60)(39,61,54,62)(43,63,48,64)

Polycyclic presentation:

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

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, 419833);