#479676

order	512
nilpotency class	3
isoclinism	isoclinic class of #479675
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17224
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

817 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)(3,8,15,31)(4,17)(5,21)(6,24)(7,11)(9,19)(10,22)(12,28)(13,29,43,33)(14,45,47,59)(16,39,25,32)(18,51)(20,35)(23,41)(26,40)(27,38)(30,50,36,48)(34,54)(37,53)(42,61,46,64)(44,58,49,57)(52,55)(56,63,60,62);(1,3)(2,8,11,31)(4,13)(5,14)(6,25)(7,15)(9,33,17,29)(10,36,38,30)(12,39,24,32)(16,28)(18,46)(19,43)(20,49)(21,45,41,59)(22,50)(23,47)(26,44)(27,48)(34,60,37,56)(35,57,40,58)(42,52)(51,64,55,61)(53,62)(54,63);(1,4)(2,9,11,17)(3,13)(5,18)(6,26)(7,19)(8,29,31,33)(10,37,38,34)(12,40,24,35)(14,46)(15,43)(16,49)(20,28)(21,55,41,51)(22,54)(23,52)(25,44)(27,53)(30,56,36,60)(32,58,39,57)(42,47)(45,64,59,61)(48,62)(50,63);(1,5,7,23)(2,10,11,38)(3,14,15,47)(4,18,19,52)(6,27,28,22)(8,30,31,36)(9,34,17,37)(12,41,24,21)(13,42,43,46)(16,50,25,48)(20,54,26,53)(29,56,33,60)(32,59,39,45)(35,51,40,55)(44,63,49,62)(57,64,58,61)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285398, #72286082, #72286106
maximal normal subgroup of	#72285398, #72286082, #72286106

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386872);