#483026

order	512
nilpotency class	4
isoclinism	isoclinic class of #483025
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17675
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#97	C₂⁴
prime radical	#60717	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ #97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

654 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=h,e^2=h,f^2=g,g^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,d^c=di,f^c=fi,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390222);