#396522

order	512
nilpotency class	3
isoclinism	isoclinic class of #393998
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14734
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1819 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:

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

Polycyclic presentation:

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

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, 303718);