#470604

order	512
nilpotency class	3
isoclinism	isoclinic class of #470603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16690
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

856 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:

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

Polycyclic presentation:

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

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, 377800);