#8741910

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

4104 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:

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

Polycyclic presentation:

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

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, 8649106);