#503499

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	#62327	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 747 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)(3,17)(4,22,29,54)(5,30)(6,15)(7,14)(8,13)(9,16)(10,19)(11,27,45,61)(12,20)(18,42,50,51)(21,46)(23,55,58,63)(24,48)(25,36,57,64)(26,60,28,53)(31,47)(32,38)(33,37)(34,39)(35,41)(40,49,52,59)(43,62,44,56);(1,3,6,19,9,21,8,34)(2,10,13,17,16,39,15,46)(4,23,26,50,29,58,28,18)(5,7,20,24,32,33,31,35)(11,40,43,64,45,52,44,36)(12,37,30,48,47,14,38,41)(22,51,53,55,54,42,60,63)(25,56,49,61,57,62,59,27);(1,4,24,56)(2,11,41,53)(3,18,32,57)(5,25,21,50)(6,26,33,61)(7,27,8,28)(9,29,35,62)(10,36,12,42)(13,43,37,54)(14,22,15,44)(16,45,48,60)(17,40,30,63)(19,23,31,59)(20,49,34,58)(38,55,46,52)(39,64,47,51);(1,5,6,20,9,32,8,31)(2,12,13,30,16,47,15,38)(3,7,19,24,21,33,34,35)(4,25,26,49,29,57,28,59)(10,37,17,48,39,14,46,41)(11,42,43,63,45,51,44,55)(18,27,23,56,50,61,58,62)(22,52,53,36,54,40,60,64)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=ef,d^2=e,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, 410695);