#494842

order	512
nilpotency class	3
isoclinism	isoclinic class of #11115
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#388
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1924
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

733 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

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, 402038);