#483314

order	512
nilpotency class	4
isoclinism	isoclinic class of #483313
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17754
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3512	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #2875 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

647 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:

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

Polycyclic presentation:

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

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, 390510);