#478254

order	512
nilpotency class	3
isoclinism	isoclinic class of #478253
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17137
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

921 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)(3,8,15,30)(4,9)(5,21)(6,24)(7,11)(10,23)(12,27)(13,41,43,28)(14,29,45,36)(16,39,25,31)(17,32,49,59)(18,33)(19,52)(20,34)(22,38)(26,40)(35,54)(37,51)(42,55,44,63)(46,57,47,56)(48,60,64,58)(50,62)(53,61);(1,3)(2,8,11,30)(4,17)(5,14)(6,25)(7,15)(9,32,34,59)(10,36,21,29)(12,39,24,31)(13,19)(16,27)(18,48)(20,49)(22,47)(23,45)(26,46)(28,52,41,35)(33,60,37,58)(38,56,40,57)(42,53)(43,54)(44,50)(51,64)(55,62,63,61);(1,4)(2,9)(3,13,15,43)(5,18)(6,19)(7,20)(8,28,30,41)(10,37)(11,34)(12,35)(14,44,45,42)(16,17,25,49)(21,33)(22,50)(23,51)(24,52)(26,53)(27,54)(29,55,36,63)(31,32,39,59)(38,62)(40,61)(46,48,47,64)(56,60,57,58);(1,5,7,23)(2,10)(3,14,15,45)(4,18,20,51)(6,26,27,22)(8,29)(9,33)(11,21)(12,40)(13,42,43,44)(16,47,25,46)(17,48,49,64)(19,53,54,50)(24,38)(28,55)(30,36)(31,57)(32,58)(34,37)(35,62)(39,56)(41,63)(52,61)(59,60)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=f,d^2=efi,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,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	#72319592, #72319962, #72320088
maximal normal subgroup of	#72319592, #72319962, #72320088

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385450);