#9498343

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3917 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=ghi,d^2=gh,e^2=h,f^2=h,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, 9405539);