#510185

order	512
nilpotency class	4
isoclinism	isoclinic class of #510185
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31374
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3067	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

407 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=f,c^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gh,h^b=hi,g^c=ghi,h^c=hi,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, 417381);