#8284487

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	#58880	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4090 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8191683);