#501370

order	512
nilpotency class	4
isoclinism	isoclinic class of #501370
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24531
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#576	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

640 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

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

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, 408566);