#422605

order	512
nilpotency class	3
isoclinism	isoclinic class of #418470
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2676
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15308
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

970 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)(3,14,18,34)(4,10)(5,24)(6,12)(7,29)(8,13)(9,17,33,28)(11,26)(15,42,19,41)(16,45,46,32)(20,50)(21,36)(22,37)(23,54)(25,56,47,38)(27,39)(30,40)(31,49,59,44)(35,52)(43,58,48,64)(51,63,57,60)(53,61)(55,62);(1,3,6,28,8,18,7,17)(2,9,12,34,13,33,29,14)(4,19,21,44,22,15,23,49)(5,25,26,16,27,47,30,46)(10,31,36,42,37,59,54,41)(11,38,39,32,40,56,24,45)(20,51,52,48,53,57,55,43)(35,60,61,58,62,63,50,64);(1,4,6,21,8,22,7,23)(2,10,12,36,13,37,29,54)(3,15,17,44,18,19,28,49)(5,20,26,52,27,53,30,55)(9,31,14,41,33,59,34,42)(11,35,39,61,40,62,24,50)(16,43,25,57,46,48,47,51)(32,58,38,60,45,64,56,63);(1,5,6,26,8,27,7,30)(2,11,12,39,13,40,29,24)(3,16,17,25,18,46,28,47)(4,20,21,52,22,53,23,55)(9,32,14,38,33,45,34,56)(10,35,36,61,37,62,54,50)(15,43,44,57,19,48,49,51)(31,58,41,60,59,64,42,63)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=e,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=gi,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, 329801);