#9508566

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3715 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,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, 9415762);