#8837071

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

4111 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=ghi,d^2=gh,e^2=gh,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, 8744267);