#395824

order	512
nilpotency class	3
isoclinism	isoclinic class of #394481
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2472
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14727
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1753 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303020);