#396532

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

1827 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)(4,17,19,35)(5,21,22,10)(6,11)(7,12)(9,20,33,24)(13,38,40,47)(14,28,42,52)(15,29)(16,30)(18,48)(23,36,25,34)(26,37)(27,41,31,44)(32,49)(39,61)(43,55,45,56)(46,57)(50,59)(51,53)(54,64)(58,62)(60,63);(1,3,6,15,7,16,26,46)(2,8,11,29,12,30,37,57)(4,13,24,44,19,40,20,41)(5,14,25,45,22,42,23,43)(9,31,35,38,33,27,17,47)(10,28,36,56,21,52,34,55)(18,39,49,62,50,63,51,64)(32,58,59,60,53,54,48,61);(1,4)(2,9)(3,13)(5,18)(6,24)(7,19)(8,27)(10,32)(11,35)(12,33)(14,39)(15,44)(16,40)(17,37)(20,26)(21,53)(22,50)(23,51)(25,49)(28,54)(29,47)(30,31)(34,48)(36,59)(38,57)(41,46)(42,63)(43,64)(45,62)(52,58)(55,60)(56,61);(1,5)(2,10)(3,14)(4,18)(6,25)(7,22)(8,28)(9,32)(11,36)(12,21)(13,39)(15,45)(16,42)(17,48)(19,50)(20,51)(23,26)(24,49)(27,54)(29,56)(30,52)(31,58)(33,53)(34,37)(35,59)(38,60)(40,63)(41,64)(43,46)(44,62)(47,61)(55,57)

Polycyclic presentation:

2^9|a^2=e,b^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,h^b=hi,e^c=ei,f^c=fi,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, 303728);