#421526

order	512
nilpotency class	3
isoclinism	isoclinic class of #421157
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2610
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15293
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

924 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)(4,19)(5,25)(6,30)(7,12)(8,13)(9,17)(10,23)(11,29)(15,36)(16,39)(18,34)(20,47)(21,37)(22,38)(24,46)(26,33)(27,40)(28,55)(31,35)(32,49)(41,43)(42,59)(44,54)(45,52)(48,62)(50,61)(51,63)(53,60)(56,58)(57,64);(1,3)(2,9)(4,20,23,49)(5,26)(6,31)(7,17)(8,18)(10,36,19,41)(11,39)(12,34)(13,35)(14,30)(15,24,43,22)(16,27)(21,50,52,42)(25,54)(28,44)(29,56)(32,38,47,46)(33,40)(37,57,45,62)(48,53,64,51)(55,58)(59,63,61,60);(1,4,6,22,8,23,7,24)(2,10,12,38,13,19,30,46)(3,15,17,20,18,43,31,49)(5,21,27,51,29,52,28,53)(9,32,14,41,35,47,34,36)(11,37,40,60,25,45,55,63)(16,42,26,48,44,50,56,64)(33,57,39,61,58,62,54,59);(1,5,6,27,8,29,7,28)(2,11,12,40,13,25,30,55)(3,16,17,26,18,44,31,56)(4,21,22,51,23,52,24,53)(9,33,14,39,35,58,34,54)(10,37,38,60,19,45,46,63)(15,42,20,48,43,50,49,64)(32,57,41,61,47,62,36,59)

Polycyclic presentation:

2^9|b^2=f,c^2=e,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,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, 328722);