#506354

order	512
nilpotency class	4
isoclinism	isoclinic class of #500306
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3227
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#28202
prime radical	#3150	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

517 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 142 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)(3,17)(4,24,27,58)(5,28)(6,16)(7,31)(8,13)(9,15)(10,20)(11,25,42,60)(12,29)(14,30)(18,48,52,63)(19,47)(21,37)(22,38)(23,39)(26,44,34,57)(32,43)(33,45,59,41)(35,61,40,54)(36,53)(46,51,64,49)(50,56,55,62);(1,3)(2,10,15,38)(4,18)(5,19)(6,20)(7,32)(8,22)(9,23)(11,40,42,35)(12,36,31,47)(13,17,16,39)(14,43,28,37)(21,29)(24,56,58,62)(25,49)(26,50)(27,52)(30,53)(33,61)(34,55)(41,64,45,46)(44,63,57,48)(51,60)(54,59);(1,4)(2,11)(3,18)(5,25)(6,26)(7,33)(8,34)(9,27)(10,35)(12,24)(13,41)(14,44)(15,42)(16,45)(17,46)(19,49)(20,50)(21,54)(22,55)(23,52)(28,57)(29,59)(30,60)(31,58)(32,61)(36,62)(37,63)(38,40)(39,64)(43,48)(47,56)(51,53);(1,5,6,29,9,30,8,7)(2,12,13,28,15,31,16,14)(3,19,20,21,23,53,22,32)(4,25,26,59,27,60,34,33)(10,36,17,37,38,47,39,43)(11,24,41,57,42,58,45,44)(18,49,50,54,52,51,55,61)(35,62,46,63,40,56,64,48)

Polycyclic presentation:

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

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, 413550);