#9058283

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	#2403	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4191 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,15)(14,16)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,45)(46,48);(2,7)(3,12)(9,16)(10,13)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,31)(30,32)(33,34,38,42)(35,39,40,36)(37,41,46,48)(43,47,45,44);(2,7)(3,12)(4,14)(5,15)(18,22)(19,26)(20,29)(31,32)(33,35,38,40)(34,39,42,36)(37,45,46,43)(41,44,48,47);(1,3)(2,8)(4,10)(5,9)(6,12)(7,11)(13,14)(15,16)(17,19)(18,23)(20,30)(21,26)(22,25)(24,32)(27,29)(28,31)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,46)(47,48);(1,4)(2,9)(3,13)(5,11)(6,14)(7,16)(8,15)(10,12)(17,20)(18,24)(19,27)(21,29)(22,28)(23,31)(25,32)(26,30)(33,36)(34,40)(35,42)(37,44)(38,39)(41,45)(43,48)(46,47);(1,5,6,15)(2,10,7,13)(3,9,12,16)(4,11,14,8)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,37,38,46)(34,41,42,48)(35,43,40,45)(36,44,39,47)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965479);