#427265

order	512
nilpotency class	3
isoclinism	isoclinic class of #418498
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2673
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15368
prime radical	#23414	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1309 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)(3,9)(4,10)(5,23)(6,12)(7,29)(8,13)(11,27)(14,43)(15,46)(16,32)(17,33)(18,41)(19,50)(20,52)(21,36)(22,37)(24,56)(25,42)(26,40)(28,38)(30,45)(31,48)(34,49)(35,53)(39,47)(44,63)(51,58)(54,60)(55,61)(57,59)(62,64);(1,3)(2,9)(4,19)(5,24)(6,18)(7,17)(8,16)(10,34)(11,39)(12,41)(13,32)(14,22)(15,25)(20,51)(21,49)(23,46)(26,47)(27,48)(28,45)(29,33)(30,37)(31,40)(35,59)(36,50)(38,43)(42,56)(44,55)(52,58)(53,57)(54,64)(60,62)(61,63);(1,4)(2,10)(3,14)(5,20)(6,28)(7,22)(8,21)(9,30)(11,35)(12,38)(13,36)(15,44)(16,45)(17,49)(18,19)(23,52)(24,57)(25,54)(26,55)(27,53)(29,37)(31,58)(32,43)(33,50)(34,41)(39,62)(40,61)(42,60)(46,59)(47,51)(48,64)(56,63);(1,5)(2,11)(3,15,16,24)(4,20)(6,27)(7,26)(8,25)(9,31,32,39)(10,35)(12,42)(13,40)(14,44,45,57)(17,48,18,47)(19,51,49,64)(21,54)(22,55)(23,29)(28,53)(30,58,43,62)(33,56,41,46)(34,59,50,63)(36,61)(37,52)(38,60)

Polycyclic presentation:

2^9|d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=ei,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, 334461);