#8393305

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) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8300501);