#510804

order	512
nilpotency class	4
isoclinism	isoclinic class of #510804
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31945
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#472	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

418 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)(3,16)(4,23,28,11)(5,13)(6,14)(7,12)(8,15)(9,19)(10,26,42,25)(17,52,56,35)(18,43,57,61)(20,39)(21,51)(22,38)(24,36,59,40)(27,48,32,44)(29,46)(30,37)(31,41,53,64)(33,45)(34,49)(47,54)(50,55,62,58)(60,63);(1,3,5,19,8,22,7,20)(2,9,12,16,15,39,13,38)(4,24,26,18,28,59,25,57)(6,30,29,54,34,60,33,21)(10,40,11,43,42,36,23,61)(14,47,45,37,49,51,46,63)(17,53,58,32,56,31,55,27)(35,44,50,64,52,48,62,41);(1,4)(2,10,15,42)(3,17,22,56)(5,26)(6,31)(7,25)(8,28)(9,35)(11,13,23,12)(14,44,49,48)(16,50)(18,54,57,21)(19,58,20,55)(24,30,59,60)(27,29)(32,33)(34,53)(36,47)(37,61)(38,62)(39,52)(40,51)(41,45,64,46)(43,63);(1,4)(2,11,15,23)(3,18,22,57)(5,25)(6,32)(7,26)(8,28)(9,36)(10,13,42,12)(14,48,49,44)(16,43)(17,54,56,21)(19,24,20,59)(27,34)(29,53)(30,55,60,58)(31,33)(35,47)(37,62)(38,61)(39,40)(41,45,64,46)(50,63)(51,52)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418000);