#479171

order	512
nilpotency class	3
isoclinism	isoclinic class of #478348
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2754
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17189
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

637 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386367);