#510898

order	512
nilpotency class	4
isoclinism	isoclinic class of #510898
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32028
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28496	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

491 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)(3,15)(4,21)(5,13)(6,12)(7,14)(8,17)(9,25)(10,20)(11,23)(16,32)(18,42)(19,34)(22,35)(24,39)(26,43)(27,37)(28,44)(29,59)(30,60)(31,45)(33,50)(36,47)(38,57)(40,52)(41,56)(46,48)(49,62)(51,63)(53,54)(55,64)(58,61);(1,3,5,17,7,20,23,19)(2,8,11,15,14,34,13,10)(4,22,24,54,27,55,25,51)(6,29,28,18,31,50,56,52)(9,35,37,63,39,64,21,53)(12,42,41,59,45,40,44,33)(16,30,47,58,49,57,48,26)(32,61,46,60,62,43,36,38);(1,4,23,25,7,27,5,24)(2,9,11,37,14,39,13,21)(3,16,17,47,20,49,19,48)(6,30,56,26,31,57,28,58)(8,32,10,36,34,62,15,46)(12,43,41,38,45,61,44,60)(18,51,50,22,52,54,29,55)(33,63,40,35,59,53,42,64);(1,3)(2,10)(4,16)(5,19)(6,18)(7,20)(8,13)(9,36)(11,34)(12,40)(14,15)(17,23)(21,32)(22,26)(24,48)(25,47)(27,49)(28,29)(30,51)(31,52)(33,44)(35,43)(37,62)(38,53)(39,46)(41,59)(42,45)(50,56)(54,57)(55,58)(60,63)(61,64)

Polycyclic presentation:

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

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, 418094);