#471228

order	512
nilpotency class	3
isoclinism	isoclinic class of #459756
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2761
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16779
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

658 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378424);