#9143382

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3946 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9050578);