#445067

order	512
nilpotency class	3
isoclinism	isoclinic class of #421157
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2610
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15800
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

680 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=f,c^2=eg,d^2=e,e^2=i,f^2=i,g^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,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, 352263);