#503502

order	512
nilpotency class	4
isoclinism	isoclinic class of #503499
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25900
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

735 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 735 conjugacy classes of subgroups — too many to draw. Only its 130 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410698);