#9058768

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	#46478	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4059 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)(4,9)(6,7)(8,10)(11,14)(12,15)(13,16)(17,18)(19,23)(20,25)(21,29)(22,24)(26,30)(27,32)(28,31)(33,34)(35,43)(36,47)(37,42)(38,39)(40,46)(41,45)(44,48);(2,7)(3,10)(4,12)(14,16)(17,19,22,20)(18,23,24,25)(21,27,30,28)(26,31,29,32)(33,35,39,46)(34,40,38,43)(36,37,45,44)(41,42,47,48);(3,10)(4,12)(5,8)(9,15)(18,24)(19,20)(21,30)(31,32)(34,38)(36,45)(37,44)(40,43);(1,3)(2,8)(4,13)(5,7)(6,10)(9,14)(11,12)(15,16)(17,18)(19,25)(20,23)(21,26)(22,24)(27,32)(28,31)(29,30)(33,36)(34,41)(35,44)(37,46)(38,47)(39,45)(40,48)(42,43);(1,4)(2,9)(3,11)(5,14)(6,12)(7,15)(8,16)(10,13)(17,20,22,19)(18,25,24,23)(21,28,30,27)(26,32,29,31)(33,37,39,44)(34,42,38,48)(35,45,46,36)(40,47,43,41);(1,5,6,8)(2,3,7,10)(4,14,12,16)(9,11,15,13)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,29)(25,32)(33,38,39,34)(35,43,46,40)(36,47,45,41)(37,48,44,42)

Polycyclic presentation:

2^9|b^2=h,e^2=h,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, 8965964);