#8360922

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4203 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:

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

Polycyclic presentation:

2^9|a^2=hi,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, 8268118);