#8407050

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

4161 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)(3,8)(4,14)(5,9)(6,10)(7,11)(12,16)(13,15)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32)(33,34)(35,39)(36,42)(37,43)(38,41)(40,44)(45,48)(46,47);(1,3,7,12)(2,8,11,16)(4,15,9,6)(5,10,14,13)(18,23)(19,26)(25,28)(27,32)(33,35)(34,41)(36,45)(37,46)(38,40)(39,44)(42,47)(43,48);(3,12)(6,15)(8,16)(10,13)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,36,40,37)(34,42,44,43)(35,45,38,46)(39,48,41,47);(1,4)(2,5)(3,6)(7,9)(8,13)(10,16)(11,14)(12,15)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,37,40,36)(34,43,44,42)(35,45,38,46)(39,48,41,47);(1,5)(2,9)(3,13)(4,11)(6,16)(7,14)(8,15)(10,12)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,38,40,35)(34,41,44,39)(36,46,37,45)(42,47,43,48);(1,6)(2,10)(3,4)(5,16)(7,15)(8,14)(9,12)(11,13)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)(33,39,40,41)(34,35,44,38)(36,47,37,48)(42,46,43,45)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8314246);