#507731

order	512
nilpotency class	4
isoclinism	isoclinic class of #499953
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3205
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#29364
prime radical	#13340	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

525 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 525 conjugacy classes of subgroups — too many to draw. Only its 124 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)(3,15)(4,21)(5,11)(6,26)(7,12)(8,14)(9,18)(10,23)(13,29)(16,46)(17,35)(19,33)(20,34)(22,55)(24,56)(25,43)(27,41)(28,37)(30,40)(31,42)(32,51)(36,59)(38,60)(39,58)(44,49)(45,52)(47,64)(48,54)(50,61)(53,63)(57,62);(1,3,5,18,8,17,7,20)(2,9,12,15,14,34,11,35)(4,16,22,48,24,47,31,53)(6,27,25,52,30,19,29,50)(10,36,38,62,39,32,37,44)(13,41,40,61,43,33,26,45)(21,54,42,46,56,63,55,64)(23,57,28,59,58,49,60,51);(1,4)(2,10)(3,16)(5,22)(6,28)(7,31)(8,24)(9,32)(11,37)(12,38)(13,42)(14,39)(15,44)(17,47)(18,48)(19,51)(20,53)(21,26)(23,29)(25,58)(27,59)(30,60)(33,46)(34,36)(35,62)(40,56)(41,64)(43,55)(45,63)(49,52)(50,57)(54,61);(2,11,14,12)(3,17)(9,15,34,35)(10,37,39,38)(13,26,43,40)(16,47)(18,20)(19,27)(21,55,56,42)(32,44,36,62)(33,45,41,61)(46,63,64,54)(48,53)(49,57)(50,52)(51,59)

Polycyclic presentation:

2^9|b^2=e,d^2=g,e^2=i,f^2=hi,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,d^b=di,f^b=fi,f^c=fh,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, 414927);