#8829220

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4168 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8736416);