#512638

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	C₄⋊Q₈	C₂⁴
prime radical	#3499	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×D₄ ◁ C₂×#41 ◁ #3499 ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 784 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,e^2=hi,f^2=i,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, 419834);