#8712126

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

4022 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8619322);