#478212

order	512
nilpotency class	3
isoclinism	isoclinic class of #478211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17136
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

843 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 843 conjugacy classes of subgroups — too many to draw. Only its 168 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=efi,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=def,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei

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, 385408);