#474325

order	512
nilpotency class	3
isoclinism	isoclinic class of #455699
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2797
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16955
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

635 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381521);