#427843

order	512
nilpotency class	3
isoclinism	isoclinic class of #427842
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	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1184 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)(4,20)(5,26)(6,13)(7,12)(8,14)(10,24)(11,29)(15,43)(16,47)(17,34)(18,33)(19,35)(21,31)(22,37)(23,39)(25,38)(27,56)(28,42)(30,41)(32,48)(36,45)(40,49)(44,58)(46,50)(51,53)(52,59)(54,61)(55,60)(57,62)(63,64);(1,3,8,19)(2,9,14,35)(4,21,23,46)(5,27,28,16)(6,18,7,17)(10,36,38,43)(11,40,41,32)(12,34,13,33)(15,25,45,24)(20,50,39,31)(22,52,54,64)(26,47,42,56)(29,48,30,49)(37,59,61,63)(44,53,57,55)(51,62,60,58);(1,4)(2,10)(3,15)(5,22)(6,25)(7,24)(8,23)(9,31)(11,37)(12,39)(13,20)(14,38)(16,44)(17,46)(18,21)(19,45)(26,51)(27,57)(28,54)(29,55)(30,53)(32,58)(33,36)(34,43)(35,50)(40,62)(41,61)(42,60)(47,63)(48,52)(49,64)(56,59);(1,5,6,30,8,28,7,29)(2,11,12,42,14,41,13,26)(3,16,17,49,19,27,18,48)(4,22,25,53,23,54,24,55)(9,32,33,56,35,40,34,47)(10,37,39,60,38,61,20,51)(15,44,46,64,45,57,21,52)(31,58,36,59,50,62,43,63)

Polycyclic presentation:

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

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, 335039);