#9171663

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	#57213	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4339 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9078859);