#508807

order	512
nilpotency class	4
isoclinism	isoclinic class of #508805
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#30137
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1747 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

417 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416003);