#8583471

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3974 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8490667);