#9054418

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	#41976	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3919 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)(3,7)(4,11)(5,13)(6,8)(9,14)(10,16)(12,15)(17,18)(19,21)(20,24)(22,26)(23,25)(27,29)(28,30)(31,32)(33,34)(35,36)(37,45)(38,39)(40,43)(41,46)(42,47)(44,48);(1,3,6,4)(2,7,8,11)(5,10,14,12)(9,15,13,16)(17,19,22,25)(18,23,26,21)(20,27,30,32)(24,31,28,29)(34,39)(36,40)(41,45)(44,47);(2,8)(3,4)(5,14)(15,16)(17,18,22,26)(19,23,25,21)(20,28,30,24)(27,29,32,31)(33,34,38,39)(35,40,43,36)(37,41,46,45)(42,47,48,44);(1,4,6,3)(2,7,8,11)(5,12,14,10)(9,15,13,16)(17,18)(19,23)(20,28)(21,25)(22,26)(24,30)(27,29)(31,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,45)(46,48);(1,2)(3,7)(4,11)(5,9)(6,8)(10,15)(12,16)(13,14)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28)(33,36,38,40)(34,35,39,43)(37,44,46,47)(41,42,45,48);(1,5,6,14)(2,9,8,13)(3,10,4,12)(7,15,11,16)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,32,28,27)(33,37,38,46)(34,41,39,45)(35,42,43,48)(36,44,40,47)

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8961614);