#507265

order	512
nilpotency class	4
isoclinism	isoclinic class of #507264
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#28984
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3067	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1765 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

445 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)(4,22)(5,11)(6,29)(7,12)(8,14)(9,18)(10,25)(13,16)(17,48)(19,34)(20,54)(21,35)(23,37)(24,40)(26,59)(27,38)(28,42)(30,46)(31,39)(32,41)(33,50)(36,53)(43,51)(44,60)(45,49)(47,64)(52,61)(55,56)(57,62)(58,63);(1,3,16,20,5,18,32,51,8,21,6,30,7,19,28,53)(2,9,29,36,11,35,42,54,14,34,13,43,12,15,41,46)(4,23,26,49,25,57,47,17,27,58,60,52,24,55,31,50)(10,37,39,61,38,62,59,33,40,63,64,45,22,56,44,48);(1,4,8,27)(2,10,14,40)(3,17,21,50)(5,24,7,25)(6,31,16,47)(9,33,34,48)(11,22,12,38)(13,44,29,59)(15,45,35,61)(18,49,19,52)(20,55,30,57)(23,51,58,53)(26,28,60,32)(36,56,43,62)(37,54,63,46)(39,41,64,42);(2,11,14,12)(3,18,21,19)(6,16)(9,34)(10,22,40,38)(13,42,29,41)(15,35)(17,49,50,52)(20,53,30,51)(23,57,58,55)(26,60)(28,32)(31,47)(33,48)(39,59,64,44)(45,61)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414461);