#8838586

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

4301 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)(3,9,10,7)(4,12,14,8)(11,15,13,16)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,28)(29,32)(33,34,38,39)(35,42,40,36)(37,45,46,41)(43,44,47,48);(2,5)(7,9)(8,12)(15,16)(17,19)(18,23)(20,21)(22,27)(24,30)(25,32)(26,28)(29,31)(34,39)(36,42)(37,46)(43,47);(3,10)(7,9)(11,13)(15,16)(19,27)(20,29)(23,28)(24,32)(33,35,38,40)(34,36,39,42)(37,43,46,47)(41,44,45,48);(1,3,6,10)(2,7,5,9)(4,13,14,11)(8,16,12,15)(17,20,22,29)(18,24,26,32)(19,21,27,31)(23,30,28,25)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47);(1,4)(2,8)(3,11)(5,12)(6,14)(7,15)(9,16)(10,13)(17,21)(18,25)(19,20)(22,31)(23,32)(24,28)(26,30)(27,29)(33,36)(34,40)(35,39)(37,44)(38,42)(41,47)(43,45)(46,48);(1,5,6,2)(3,9,10,7)(4,12,14,8)(11,16,13,15)(17,18)(19,28)(20,24)(21,25)(22,26)(23,27)(29,32)(30,31)(33,37)(34,41)(35,43)(36,44)(38,46)(39,45)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=ghi,f^2=gh,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, 8745782);