#9016503

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	#49108	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3925 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)(3,8)(4,10)(5,11)(6,16)(7,9)(12,15)(13,14)(17,18,23,21)(19,20,26,24)(22,29,30,25)(27,31,32,28);(1,3,7,6)(2,8,9,16)(4,12,14,11)(5,10,15,13)(18,21)(19,26)(25,29)(27,32)(33,34,39,43)(35,44,46,40)(36,38,42,48)(37,41,45,47);(2,9)(3,6)(4,14)(5,15)(17,19,23,26)(18,24,21,20)(22,27,30,32)(25,31,29,28)(33,35,39,46)(34,40,43,44)(36,37,42,45)(38,47,48,41);(1,4)(2,10)(3,12)(5,16)(6,11)(7,14)(8,15)(9,13)(17,20)(18,19)(21,26)(22,28)(23,24)(25,27)(29,32)(30,31)(33,36,39,42)(34,38,43,48)(35,45,46,37)(40,41,44,47);(1,5,7,15)(2,11,9,12)(3,13,6,10)(4,8,14,16)(17,21,23,18)(19,20,26,24)(22,29,30,25)(27,28,32,31)(33,37,39,45)(34,41,43,47)(35,36,46,42)(38,40,48,44);(1,6,7,3)(2,8,9,16)(4,11,14,12)(5,10,15,13)(17,22,23,30)(18,25,21,29)(19,27,26,32)(20,28,24,31)(33,38)(34,42)(35,41)(36,43)(37,40)(39,48)(44,45)(46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8923699);