#149422

order	512
nilpotency class	3
isoclinism	isoclinic class of #2032
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#393
commutator, prime radical	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11380
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

665 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 665 conjugacy classes of subgroups — too many to draw. Only its 204 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,g^2=h,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=di,d^c=di,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56618);