#453834

order	512
nilpotency class	3
isoclinism	isoclinic class of #429607
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2595
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16145
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

527 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361030);