#9508569

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

4011 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)(3,5)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(33,34,39,38)(35,41,36,40)(37,42,46,45)(43,48,44,47);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(17,18,22,23)(19,21,25,30)(20,24,27,31)(26,29,28,32)(33,35,39,36)(34,40,38,41)(37,43,46,44)(42,47,45,48);(2,4)(3,10)(9,12)(11,16)(18,23)(20,27)(21,30)(26,28)(33,35,39,36)(34,41,38,40)(37,44,46,43)(42,47,45,48);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,19,22,25)(18,21,23,30)(20,28,27,26)(24,32,31,29)(33,36,39,35)(34,41,38,40)(37,43,46,44)(42,47,45,48);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,37,39,46)(34,42,38,45)(35,43,36,44)(40,47,41,48);(1,6,7,14)(2,9,4,12)(3,11,10,16)(5,13,8,15)(17,21)(18,25)(19,23)(20,29)(22,30)(24,28)(26,31)(27,32)(33,38,39,34)(35,41,36,40)(37,45,46,42)(43,48,44,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9415765);