#443360

order	512
nilpotency class	3
isoclinism	isoclinic class of #443359
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15757
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

673 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 350556);