#428197

order	512
nilpotency class	3
isoclinism	isoclinic class of #423438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2712
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15379
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

918 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)(4,21)(5,27)(6,13)(7,12)(8,14)(9,18)(10,24)(11,29)(16,32)(17,47)(19,35)(20,34)(22,36)(23,53)(25,39)(26,38)(28,40)(30,41)(31,42)(33,49)(37,56)(43,45)(44,62)(46,50)(48,59)(51,55)(52,58)(54,60)(57,61)(63,64);(1,3)(2,9)(4,22)(5,28)(6,19)(7,18)(8,20)(10,36)(11,40)(12,34)(13,15)(14,35)(16,24)(17,29)(21,50)(23,54)(25,46)(26,45)(27,33)(30,48)(31,49)(32,38)(37,58)(39,43)(41,47)(42,59)(44,56)(51,60)(52,55)(53,64)(57,63)(61,62);(1,4,6,24,8,26,7,25)(2,10,12,21,14,39,13,38)(3,16,18,22,20,46,19,45)(5,23,29,55,30,56,31,57)(9,32,15,43,35,50,34,36)(11,37,27,51,42,53,41,61)(17,44,28,52,49,54,48,63)(33,58,40,62,47,64,59,60);(1,5,8,30)(2,11,14,42)(3,17)(4,23,26,56)(6,29,7,31)(9,33)(10,37,39,53)(12,27,13,41)(15,40)(16,44)(18,28)(19,48)(20,49)(21,51,38,61)(22,52)(24,55,25,57)(32,58)(34,59)(35,47)(36,60)(43,62)(45,63)(46,54)(50,64)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335393);