#501631

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	#33595	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

637 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=e,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, 408827);