#481730

order	512
nilpotency class	4
isoclinism	isoclinic class of #481729
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1505
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#3054	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

744 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)(3,14,17,43)(4,20,21,54)(5,25,28,41)(6,12)(7,31)(8,13)(9,16)(10,23,35,24)(11,29,39,30)(15,44)(18,37,19,52)(22,38)(26,40)(27,36)(32,56,61,60)(33,50)(34,51)(42,48)(45,64)(46,53,49,63)(47,62)(55,58)(57,59);(1,3,8,17)(2,9,13,34)(4,15,21,45)(5,26,28,50)(6,19,7,18)(10,32,35,61)(11,37,39,52)(12,40,31,33)(14,41,43,25)(16,29,51,30)(20,53,54,63)(22,56,27,60)(23,48,24,47)(36,64,38,44)(42,55,62,59)(46,57,49,58);(1,4)(2,10,13,35)(3,15)(5,27)(6,23)(7,24)(8,21)(9,32,34,61)(11,38,39,36)(12,20,31,54)(14,42,43,62)(16,49)(17,45)(18,47)(19,48)(22,28)(25,59,41,55)(26,60)(29,58)(30,57)(33,63,40,53)(37,44,52,64)(46,51)(50,56);(4,21)(10,35)(15,45)(20,54)(22,27)(23,24)(32,61)(36,38)(42,62)(44,64)(46,49)(47,48)(53,63)(55,59)(56,60)(57,58)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cg,e^a=eg,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi,d^c=di,e^c=ei,f^c=fi,f^e=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, 388926);