#431131

order	512
nilpotency class	3
isoclinism	isoclinic class of #428398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2609
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15431
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

807 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)(3,15)(4,21)(5,27,31,41)(6,13)(7,12)(8,14)(9,18)(10,25)(11,29,42,30)(16,32)(17,48,49,33)(19,35)(20,34)(22,36)(23,37,55,62)(24,39)(26,38)(28,40,50,44)(43,46)(45,63,64,60)(47,51)(52,54,57,56)(53,59,58,61);(1,3,6,18,8,20,7,19)(2,9,12,15,14,35,13,34)(4,22,24,47,26,46,25,16)(5,28,29,49,31,50,30,17)(10,36,38,32,39,43,21,51)(11,40,41,48,42,44,27,33)(23,45,54,58,55,64,56,53)(37,61,52,63,62,59,57,60);(1,4,7,25,8,26,6,24)(2,10,13,21,14,39,12,38)(3,16,19,46,20,47,18,22)(5,23,29,54,31,55,30,56)(9,32,34,36,35,51,15,43)(11,37,41,52,42,62,27,57)(17,45,28,58,49,64,50,53)(33,59,40,60,48,61,44,63);(1,5,7,29,8,31,6,30)(2,11,13,41,14,42,12,27)(3,17,19,28,20,49,18,50)(4,23,25,54,26,55,24,56)(9,33,34,40,35,48,15,44)(10,37,21,52,39,62,38,57)(16,45,46,58,47,64,22,53)(32,59,36,60,51,61,43,63)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=f,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,f^a=fi,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei

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, 338327);