#9065941

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3953 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)(17,18,22,26)(19,23,24,20)(21,25,31,32)(27,29,30,28)(33,34,39,37)(35,36,43,40)(38,45,47,41)(42,46,48,44);(1,3)(2,8)(4,13)(5,6)(7,12)(9,10)(11,14)(15,16)(17,19)(18,23)(20,26)(21,27)(22,24)(25,29)(28,32)(30,31)(33,35,39,43)(34,40,37,36)(38,42,47,48)(41,46,45,44);(3,12)(5,15)(8,14)(9,13)(17,20,22,23)(18,24,26,19)(21,28,31,29)(25,30,32,27)(33,35,39,43)(34,40,37,36)(38,42,47,48)(41,46,45,44);(1,4)(2,6)(3,13)(5,8)(7,10)(9,12)(11,16)(14,15)(17,21)(18,25)(19,27)(20,28)(22,31)(23,29)(24,30)(26,32)(33,36)(34,35)(37,43)(38,46)(39,40)(41,48)(42,45)(44,47);(1,5)(2,9)(3,6)(4,14)(7,15)(8,10)(11,13)(12,16)(17,20)(18,19)(21,29)(22,23)(24,26)(25,30)(27,32)(28,31)(33,37,39,34)(35,40,43,36)(38,45,47,41)(42,46,48,44);(1,6)(2,10)(3,5)(4,11)(7,16)(8,9)(12,15)(13,14)(17,19)(18,20)(21,30)(22,24)(23,26)(25,29)(27,31)(28,32)(33,38)(34,41)(35,42)(36,44)(37,45)(39,47)(40,46)(43,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,e^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, 8973137);