#478456

order	512
nilpotency class	3
isoclinism	isoclinic class of #478455
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17149
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

954 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)(3,13,16,8)(4,9,20,33)(5,22,23,39)(6,12)(7,11)(10,24,36,26)(14,31)(15,44)(17,41,47,30)(18,28)(19,51)(21,34,25,38)(27,37)(29,45)(32,54)(35,53)(40,49)(42,61,50,64)(43,59)(46,55)(48,57)(52,60)(56,63,58,62);(1,3)(2,8)(4,18,20,14)(5,15)(6,17)(7,16)(9,31,33,28)(10,29)(11,13)(12,30)(19,42,52,50)(21,48,25,43)(22,55)(23,46)(24,45)(26,49)(27,47)(32,56,35,58)(34,59,38,57)(36,40)(37,41)(39,44)(51,64,60,61)(53,62,54,63);(1,4)(2,9,11,33)(3,14)(5,19)(6,25)(7,20)(8,28,13,31)(10,35,36,32)(12,38,37,34)(15,42)(16,18)(17,48)(21,27)(22,51,39,60)(23,52)(24,54)(26,53)(29,58,40,56)(30,59,41,57)(43,47)(44,61,55,64)(45,63)(46,50)(49,62);(1,5,7,23)(2,10,11,36)(3,15,16,46)(4,19,20,52)(6,26,27,24)(8,29,13,40)(9,32,33,35)(12,39,37,22)(14,42,18,50)(17,49,47,45)(21,54,25,53)(28,56,31,58)(30,44,41,55)(34,60,38,51)(43,63,48,62)(57,61,59,64)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285398, #72286066, #72286206
maximal normal subgroup of	#72285398, #72286066, #72286206

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385652);