#427821

order	512
nilpotency class	3
isoclinism	isoclinic class of #427820
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15375
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

1202 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)(4,9)(5,20)(6,24)(7,11)(10,23)(12,28)(13,33)(14,45)(15,31)(16,39)(17,49)(18,40)(19,35)(21,55)(22,41)(25,32)(26,36)(27,38)(29,43)(30,46)(34,52)(37,48)(42,61)(44,57)(47,58)(50,64)(51,54)(53,60)(56,63)(59,62);(1,3)(2,8)(4,13)(5,21)(6,25)(7,15)(9,33)(10,37)(11,31)(12,39)(14,22)(16,28)(17,50)(18,47)(19,43)(20,45)(23,46)(24,32)(26,44)(27,48)(29,35)(30,38)(34,56)(36,57)(40,58)(41,55)(42,51)(49,61)(52,62)(53,63)(54,64)(59,60);(1,4,7,19)(2,9,11,35)(3,13,15,43)(5,17,22,51)(6,26,28,18)(8,29,31,33)(10,34,38,60)(12,40,24,36)(14,42,21,50)(16,47,25,44)(20,54,41,49)(23,53,27,52)(30,56,37,59)(32,58,39,57)(45,61,55,64)(46,63,48,62);(1,5)(2,10)(3,14,15,21)(4,17)(6,27)(7,22)(8,30,31,37)(9,34)(11,38)(12,41)(13,42,43,50)(16,48,25,46)(18,53)(19,51)(20,24)(23,28)(26,52)(29,56,33,59)(32,55,39,45)(35,60)(36,54)(40,49)(44,63,47,62)(57,61,58,64)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335017);