#9545555

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

3995 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=ghi,e^2=ghi,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, 9452751);