#421447

order	512
nilpotency class	3
isoclinism	isoclinic class of #421447
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2474
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15291
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

920 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)(3,13)(4,18)(5,24)(6,28)(7,12)(8,16)(9,22)(10,27)(11,30)(14,31)(15,37)(17,39)(19,47)(20,49)(21,36)(23,35)(25,32)(26,38)(29,33)(34,51)(40,43)(41,59)(42,60)(44,56)(45,54)(46,53)(48,63)(50,62)(52,55)(57,64)(58,61);(1,3,7,17)(2,8,12,33)(4,19)(5,25,27,58)(6,29,30,16)(9,31)(10,37,24,54)(11,39,28,13)(14,23)(15,44,45,26)(18,40)(20,50)(21,42)(22,43)(32,56,61,38)(34,62)(35,60)(36,47)(41,52)(46,63)(48,51)(49,59)(53,57)(55,64);(1,4)(2,9)(3,14)(5,20)(6,21)(7,23)(8,31)(10,34)(11,35)(12,36)(13,40)(15,41)(16,42)(17,19)(18,28)(22,30)(24,55)(25,57)(26,51)(27,53)(29,43)(32,59)(33,47)(37,62)(38,49)(39,60)(44,52)(45,48)(46,56)(50,58)(54,64)(61,63);(1,5,6,26,7,27,30,44)(2,10,11,38,12,24,28,56)(3,15,16,25,17,45,29,58)(4,20,21,51,23,53,22,52)(8,32,13,37,33,61,39,54)(9,34,35,49,36,55,18,46)(14,41,42,57,19,48,43,50)(31,59,40,62,47,63,60,64)

Polycyclic presentation:

2^9|b^2=ef,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,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, 328643);