#483024

order	512
nilpotency class	4
isoclinism	isoclinic class of #483023
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	#3455	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #2873 ◁ G
LECS	G ▷ #97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

648 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:

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

Polycyclic presentation:

2^9|a^2=f,b^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,f^c=fi,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390220);