#8549530

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

3945 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8456726);