#512659

order	512
nilpotency class	5
isoclinism	isoclinic class of #512655
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	#124	C₂⁴
prime radical	#3499	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #3499 ◁ G
LECS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 788 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)(3,15,19,32)(4,10)(5,11)(6,28)(7,13,30,41)(8,14)(9,18,33,52)(12,29,43,27)(16,45,49,60)(17,44,51,34)(20,39,21,46)(22,53,50,57)(23,35)(24,40)(25,37,56,61)(26,38)(31,47,58,63)(36,55,42,54)(48,62,64,59);(1,3,8,21)(2,9,14,34)(4,22,26,16)(5,18,6,17)(7,19,30,20)(10,31,38,59)(11,39,28,32)(12,15,43,46)(13,44,41,33)(23,47,24,48)(25,50,56,49)(27,52,29,51)(35,60,40,57)(36,53,42,45)(37,62,61,58)(54,63,55,64);(1,4,8,26)(2,10,14,38)(3,16,21,22)(5,24,6,23)(7,25,30,56)(9,31,34,59)(11,40,28,35)(12,42,43,36)(13,37,41,61)(15,45,46,53)(17,48,18,47)(19,49,20,50)(27,55,29,54)(32,60,39,57)(33,58,44,62)(51,64,52,63);(1,5,8,6)(2,11,14,28)(3,17,21,18)(4,23,26,24)(7,27,30,29)(9,32,34,39)(10,35,38,40)(12,41,43,13)(15,44,46,33)(16,47,22,48)(19,51,20,52)(25,54,56,55)(31,57,59,60)(36,61,42,37)(45,58,53,62)(49,63,50,64)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419855);