#8359857

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3991 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,34,39,36)(35,40,42,38)(37,41,47,44)(43,48,45,46);(2,7)(3,11)(4,14)(13,16)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,35,39,42)(34,40,36,38)(37,45,47,43)(41,46,44,48);(2,7)(5,9)(8,10)(13,16)(20,28)(23,30)(26,32)(29,31)(35,42)(38,40)(43,45)(46,48);(1,3,6,11)(2,8,7,10)(4,15,14,12)(5,13,9,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,36,39,34)(35,38,42,40)(37,41,47,44)(43,48,45,46);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,16)(11,15)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,48)(42,45);(1,5,6,9)(2,4,7,14)(3,13,11,16)(8,15,10,12)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,38)(34,35)(36,42)(37,46)(39,40)(41,43)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8267053);