#501030

order	512
nilpotency class	4
isoclinism	isoclinic class of #501028
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24354
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#62636	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

976 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)(3,15,21,33)(4,22,25,38)(5,12)(6,10)(7,13)(8,14)(9,18,34,20)(11,23,35,26)(16,32,45,55)(17,31)(19,30,44,40)(24,50,52,61)(27,39)(28,36)(29,41)(37,51,59,53)(42,49)(43,60)(46,56)(47,57)(48,58)(54,63)(62,64);(1,3)(2,9)(4,17)(5,20)(6,21)(7,16)(8,18)(10,34)(11,31)(12,33)(13,40)(14,15)(19,29)(22,42)(23,49)(24,43)(25,47)(26,46)(27,44)(28,45)(30,36)(32,41)(35,57)(37,60)(38,56)(39,55)(48,51)(50,58)(52,63)(53,62)(54,59)(61,64);(2,10)(3,16)(5,8)(7,29)(9,30)(11,35)(13,39)(15,32)(17,43)(18,44)(19,20)(21,45)(23,26)(24,51)(27,28)(31,54)(33,55)(34,40)(36,41)(37,61)(42,58)(46,48)(47,63)(49,62)(50,59)(52,53)(56,64)(57,60);(1,4)(2,11)(3,17)(5,26)(6,25)(7,24)(8,23)(9,31)(10,35)(12,38)(13,37)(14,22)(15,42)(16,43)(18,49)(19,48)(20,46)(21,47)(27,53)(28,52)(29,51)(30,54)(32,58)(33,56)(34,57)(36,59)(39,61)(40,60)(41,50)(44,62)(45,63)(55,64)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408226);