#502099

order	512
nilpotency class	4
isoclinism	isoclinic class of #502097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25076
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33540	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

739 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:

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

Polycyclic presentation:

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

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, 409295);