#512619

order	512
nilpotency class	5
isoclinism	isoclinic class of #512613
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#353
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄⋊Q₈	C₂⁴
prime radical	#60648	C₂
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

1063 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 1063 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)(3,14,16,37,18,30,45,55)(4,9,20,34,19,32,46,57)(6,12,26,38,27,10,23,36)(8,22,29,25,31,44,56,17)(15,40,42,59,41,54,61,64)(21,35,51,60,47,33,49,58)(28,48,53,50,52,62,63,43);(1,3,7,18)(2,8,13,31)(4,15,19,41)(5,22,24,44)(6,25,27,17)(9,28,32,52)(10,30,12,14)(11,37,39,55)(16,23,45,26)(20,48,46,62)(21,50,47,43)(29,38,56,36)(33,54,35,40)(34,59,57,64)(42,49,61,51)(53,60,63,58);(1,4)(2,9)(3,15)(5,20)(6,21)(7,19)(8,28)(10,33)(11,34)(12,35)(13,32)(14,40)(16,42)(17,43)(18,41)(22,48)(23,49)(24,46)(25,50)(26,51)(27,47)(29,53)(30,54)(31,52)(36,58)(37,59)(38,60)(39,57)(44,62)(45,61)(55,64)(56,63);(4,19)(9,32)(15,41)(20,46)(21,47)(28,52)(33,35)(34,57)(40,54)(42,61)(43,50)(48,62)(49,51)(53,63)(58,60)(59,64)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419815);