#9016445

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	#44300	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gi,d^2=ghi,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, 8923641);