#466065

order	512
nilpotency class	3
isoclinism	isoclinic class of #466064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1004 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)(5,26)(6,13)(7,12)(8,14)(9,18)(10,24)(11,28)(16,45)(17,49)(19,34)(20,42)(22,43)(23,56)(25,37)(27,39)(29,41)(30,40)(31,38)(32,48)(33,53)(35,47)(36,58)(44,52)(46,63)(50,62)(51,59)(54,57)(55,60)(61,64);(1,3)(2,9,14,34)(4,22,25,47)(5,17)(6,18)(7,19)(8,20)(10,35)(11,33,41,49)(12,15,13,42)(16,24,48,31)(21,32)(23,55,58,50)(26,59,40,44)(27,46,57,64)(28,51)(29,52)(30,53)(36,61)(37,45)(38,43)(39,62)(54,60)(56,63);(1,4,7,31,8,25,6,24)(2,10,13,37,14,38,12,21)(3,16,19,47,20,48,18,22)(5,27,29,23,30,57,28,58)(9,32,42,35,34,45,15,43)(11,39,40,36,41,54,26,56)(17,50,52,46,53,55,51,64)(33,61,44,60,49,63,59,62);(1,5,6,29,8,30,7,28)(2,11,12,40,14,41,13,26)(3,17,18,52,20,53,19,51)(4,23,24,57,25,58,31,27)(9,33,15,44,34,49,42,59)(10,36,21,54,38,56,37,39)(16,46,22,55,48,64,47,50)(32,60,43,63,45,62,35,61)

Polycyclic presentation:

2^9|b^2=fg,c^2=ei,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,h^a=hi,c^b=cg,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373261);