#436375

order	512
nilpotency class	3
isoclinism	isoclinic class of #423438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2712
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15557
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

741 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:

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

Polycyclic presentation:

2^9|a^2=efg,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, 343571);