#7614088

order	512
nilpotency class	3
isoclinism	isoclinic class of #7535048
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3549
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	#59816
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2472 conjugacy classes of subgroups; only the 558 normal subgroups are shown.

normal subgroups only

This group has 2472 conjugacy classes of subgroups — too many to draw. Only its 558 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 — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=h,e^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7521284);