#8360915

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	#58880	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4051 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:

(2,7)(3,9)(10,14)(11,16)(17,18,22,21)(19,23,27,25)(20,28,30,24)(26,32,29,31)(33,34,39,36)(35,40,42,38)(37,41,47,44)(43,48,45,46);(1,2,6,7)(3,8,9,4)(5,10,13,14)(11,15,16,12)(17,18,22,21)(19,25,27,23)(20,28,30,24)(26,31,29,32)(33,35,39,42)(34,40,36,38)(37,45,47,43)(41,46,44,48);(5,13)(10,14)(11,16)(12,15)(18,21)(23,25)(24,28)(31,32)(35,42)(38,40)(43,45)(46,48);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,14)(13,16)(17,19)(18,23)(20,29)(21,25)(22,27)(24,32)(26,30)(28,31)(33,36,39,34)(35,38,42,40)(37,41,47,44)(43,48,45,46);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,20,22,30)(18,24,21,28)(19,26,27,29)(23,31,25,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,48)(42,45);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,21,22,18)(19,25,27,23)(20,28,30,24)(26,32,29,31)(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=gi,d^2=hi,e^2=gh,f^2=h,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, 8268111);