#501633

order	512
nilpotency class	4
isoclinism	isoclinic class of #501631
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24726
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3414	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

629 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=e,d^2=ei,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi

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, 408829);