#448813

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	#15921
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

926 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)(3,15,19,35)(4,20,25,42)(5,26)(6,12)(7,13)(8,14)(9,18,34,31)(10,23,38,24)(11,28)(16,32)(17,47,48,33)(21,36)(22,37,54,63)(27,57,49,39)(29,41)(30,40)(43,45)(44,64)(46,60)(50,55,56,53)(51,62)(52,59)(58,61);(1,3,7,18,8,19,6,31)(2,9,13,15,14,34,12,35)(4,21,23,45,25,16,24,46)(5,27,29,48,30,49,28,17)(10,36,20,43,38,32,42,60)(11,39,26,47,41,57,40,33)(22,52,53,51,54,58,55,44)(37,62,50,61,63,64,56,59);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,25)(9,32)(11,37)(12,20)(13,42)(14,38)(15,43)(17,44)(18,45)(19,21)(26,56)(27,58)(28,53)(29,55)(30,54)(31,46)(33,59)(34,36)(35,60)(39,64)(40,50)(41,63)(47,61)(48,51)(49,52)(57,62);(1,5)(2,11)(3,17,19,48)(4,22)(6,28)(7,29)(8,30)(9,33,34,47)(10,37)(12,40)(13,26)(14,41)(15,39,35,57)(16,44,21,51)(18,27,31,49)(20,50)(23,53)(24,55)(25,54)(32,59,36,61)(38,63)(42,56)(43,64,60,62)(45,58,46,52)

Polycyclic presentation:

2^9|a^2=ef,b^2=f,d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,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, 356009);