#427311

order	512
nilpotency class	3
isoclinism	isoclinic class of #427311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2471
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15368
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1301 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)(3,15)(4,10)(5,25)(6,13)(7,12)(8,14)(9,18)(11,27)(16,43)(17,46)(19,34)(20,33)(21,31)(22,51)(23,38)(24,39)(26,40)(28,41)(29,42)(30,37)(32,47)(35,45)(36,54)(44,57)(48,59)(49,58)(50,60)(52,64)(53,62)(55,61)(56,63);(1,3)(2,9)(4,21)(5,26)(6,20)(7,18)(8,19)(10,35)(11,40)(12,34)(13,15)(14,33)(16,23)(17,27)(22,52)(24,49)(25,32)(28,47)(29,48)(30,45)(31,38)(36,57)(37,58)(39,43)(41,59)(42,46)(44,55)(50,53)(51,63)(54,56)(60,61)(62,64);(1,4)(2,10)(3,16)(5,22)(6,30)(7,23)(8,24)(9,31)(11,36)(12,38)(13,37)(14,39)(15,35)(17,44)(18,49)(19,45)(20,21)(25,51)(26,56)(27,54)(28,53)(29,55)(32,57)(33,58)(34,43)(40,60)(41,62)(42,61)(46,64)(47,52)(48,50)(59,63);(1,5)(2,11)(3,17,19,47)(4,22)(6,27)(7,28)(8,29)(9,32,33,46)(10,36)(12,25)(13,42)(14,41)(15,40,34,59)(16,44,45,52)(18,26,20,48)(21,50,49,56)(23,53)(24,55)(30,54)(31,57,58,64)(35,60,43,63)(37,61)(38,51)(39,62)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334507);