#453403

order	512
nilpotency class	3
isoclinism	isoclinic class of #428398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2609
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16096
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

603 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 603 conjugacy classes of subgroups — too many to draw. Only its 176 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)(4,19)(5,11,28,40)(6,29)(7,12)(8,13)(9,17)(10,23)(15,35)(16,44,45,38)(18,33)(20,48)(21,46,53,36)(22,37)(24,47)(25,55,56,32)(26,57,27,39)(30,34)(31,50)(41,43)(42,61,51,59)(49,58,64,62)(52,60,54,63);(1,3)(2,9)(4,20,23,50)(5,25)(6,30)(7,17)(8,18)(10,35,19,41)(11,38)(12,33)(13,34)(14,29)(15,24,43,22)(16,26)(21,51,53,42)(27,45)(28,56)(31,37,48,47)(32,39)(36,58,46,62)(40,44)(49,54,64,52)(55,57)(59,63,61,60);(1,4)(2,10,13,19)(3,15,18,43)(5,21)(6,22)(7,24)(8,23)(9,31)(11,36,40,46)(12,37,29,47)(14,41)(16,42,45,51)(17,20,30,50)(25,49,56,64)(26,52)(27,54)(28,53)(32,58)(33,35)(34,48)(38,61)(39,60,57,63)(44,59)(55,62);(1,5,6,26,8,28,7,27)(2,11,12,39,13,40,29,57)(3,16,17,25,18,45,30,56)(4,21,22,52,23,53,24,54)(9,32,14,38,34,55,33,44)(10,36,37,60,19,46,47,63)(15,42,20,49,43,51,50,64)(31,58,41,61,48,62,35,59)

Polycyclic presentation:

2^9|a^2=egh,b^2=f,c^2=eg,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,f^c=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 360599);