#391097

order	512
nilpotency class	3
isoclinism	isoclinic class of #391095
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14609
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

721 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 721 conjugacy classes of subgroups — too many to draw. Only its 196 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fhi,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298293);