#9065956

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	#44161	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4023 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8973152);