#470590

order	512
nilpotency class	3
isoclinism	isoclinic class of #470589
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

867 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)(3,13)(4,18)(5,22,24,10)(6,25)(7,12)(8,16)(9,20)(11,27)(14,38)(15,29,43,36)(17,30)(19,49,51,48)(21,34)(23,33,55,53)(26,32)(28,46)(31,50,52,54)(35,41)(37,45)(39,61,42,60)(40,47)(44,59,64,58)(56,63,57,62);(1,3)(2,8,12,13)(4,14)(5,15)(6,17)(7,16)(9,28,32,35)(10,29,22,36)(11,30,25,37)(18,47,34,38)(19,39)(20,41)(21,40)(23,44)(24,43)(26,46)(27,45)(31,56,52,57)(33,58,53,59)(42,51)(48,61,49,60)(50,62)(54,63)(55,64);(1,4,7,21)(2,9,12,32)(3,14,16,40)(5,19,24,51)(6,26,27,20)(8,28,13,35)(10,31,22,52)(11,34,25,18)(15,42,43,39)(17,46,45,41)(23,54,55,50)(29,57,36,56)(30,38,37,47)(33,49,53,48)(44,62,64,63)(58,61,59,60);(1,5)(2,10)(3,15)(4,19)(6,23)(7,24)(8,29)(9,31)(11,33)(12,22)(13,36)(14,39)(16,43)(17,44)(18,48)(20,50)(21,51)(25,53)(26,54)(27,55)(28,56)(30,58)(32,52)(34,49)(35,57)(37,59)(38,60)(40,42)(41,62)(45,64)(46,63)(47,61)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286082, #72286206, #72286267
maximal normal subgroup of	#72286082, #72286206, #72286267

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377786);