#8864529

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

4184 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)(3,9)(4,12)(5,6)(7,10)(8,14)(11,15)(13,16)(20,27)(24,31)(26,28)(29,32)(33,34,38,39)(35,42,40,36)(37,45,46,41)(43,44,47,48);(2,6)(7,9)(8,12)(15,16)(17,18)(19,21)(20,24)(22,23)(25,30)(26,29)(27,31)(28,32)(34,39)(36,42)(37,46)(43,47);(3,10)(7,9)(11,13)(15,16)(18,23)(19,25)(24,31)(26,28)(33,35,38,40)(34,36,39,42)(37,43,46,47)(41,44,45,48);(1,3)(2,7)(4,13)(5,10)(6,9)(8,16)(11,14)(12,15)(17,19,22,25)(18,21,23,30)(20,28,27,26)(24,32,31,29)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47);(1,4)(2,8)(3,11)(5,14)(6,12)(7,15)(9,16)(10,13)(17,20,22,27)(18,24,23,31)(19,26,25,28)(21,29,30,32)(33,36)(34,40)(35,39)(37,44)(38,42)(41,47)(43,45)(46,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,21)(18,25)(19,23)(20,29)(22,30)(24,28)(26,31)(27,32)(33,37,38,46)(34,41,39,45)(35,43,40,47)(36,44,42,48)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8771725);