#498643

order	512
nilpotency class	3
isoclinism	isoclinic class of #24986
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#555
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23250
prime radical	C₂³	#568
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

380 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 198 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=f,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=deh,e^b=ei,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405839);