#451444

order	512
nilpotency class	3
isoclinism	isoclinic class of #418871
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2697
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15998
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

763 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)(3,13,17,8)(4,18)(5,22,24,10)(6,25)(7,11)(9,21)(12,27)(14,38,40,56)(15,34)(16,30,45,37)(19,47,49,46)(20,36)(23,35,54,52)(26,33)(28,41,31,44)(29,43)(32,50,51,53)(39,60)(42,57)(48,64)(55,62)(58,61)(59,63);(1,3,6,16,7,17,27,45)(2,8,25,37,11,13,12,30)(4,14,26,44,20,40,21,41)(5,15,23,42,24,43,54,63)(9,31,18,38,33,28,36,56)(10,34,52,57,22,29,35,59)(19,48,53,61,49,39,50,62)(32,58,46,60,51,55,47,64);(1,4,6,26,7,20,27,21)(2,9,25,18,11,33,12,36)(3,14,16,44,17,40,45,41)(5,19,23,53,24,49,54,50)(8,28,37,56,13,31,30,38)(10,32,52,46,22,51,35,47)(15,39,42,62,43,48,63,61)(29,55,59,64,34,58,57,60);(1,5)(2,10)(3,15)(4,19)(6,23)(7,24)(8,29)(9,32)(11,22)(12,35)(13,34)(14,39)(16,42)(17,43)(18,46)(20,49)(21,50)(25,52)(26,53)(27,54)(28,55)(30,57)(31,58)(33,51)(36,47)(37,59)(38,60)(40,48)(41,61)(44,62)(45,63)(56,64)

Polycyclic presentation:

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

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, 358640);