#405289

order	512
nilpotency class	3
isoclinism	isoclinic class of #405288
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14911
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2161 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312485);