#9059267

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	#55587	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4107 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,15)(14,16)(17,18,23,20)(19,24,21,25)(22,28,30,26)(27,32,29,31)(33,34,39,44)(35,45,48,40)(36,38,43,41)(37,42,46,47);(2,7)(3,12)(9,16)(10,13)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,35,39,48)(34,40,44,45)(36,37,43,46)(38,47,41,42);(2,7)(3,12)(4,14)(5,15)(18,20)(19,21)(22,30)(31,32)(36,43)(37,46)(38,41)(42,47);(1,3)(2,8)(4,10)(5,9)(6,12)(7,11)(13,14)(15,16)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32)(33,36,39,43)(34,41,44,38)(35,46,48,37)(40,47,45,42);(1,4)(2,9)(3,13)(5,11)(6,14)(7,16)(8,15)(10,12)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31)(33,37,39,46)(34,42,44,47)(35,36,48,43)(38,40,41,45);(1,5,6,15)(2,10,7,13)(3,9,12,16)(4,11,14,8)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,38,39,41)(34,43,44,36)(35,47,48,42)(37,40,46,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8966463);