#8901182

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

4133 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)(3,11)(4,9)(5,12)(6,7)(8,14)(10,16)(13,15)(17,18,22,25)(19,26,20,23)(21,24,29,31)(27,32,28,30);(2,7)(4,15)(8,11)(10,12)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,31,32)(33,34)(35,39)(36,40)(37,38)(41,47)(42,45)(43,48)(44,46);(2,7)(5,16)(8,11)(9,13)(21,29)(24,31)(27,28)(30,32)(34,38)(39,45)(40,46)(47,48);(1,3,6,14)(2,8,7,11)(4,10,15,12)(5,13,16,9)(17,18)(19,23)(20,26)(21,24)(22,25)(27,30)(28,32)(29,31)(33,35,37,42)(34,39,38,45)(36,43,44,41)(40,48,46,47);(1,4,6,15)(2,9,7,13)(3,12,14,10)(5,8,16,11)(17,20,22,19)(18,23,25,26)(21,28,29,27)(24,30,31,32)(33,36,37,44)(34,40,38,46)(35,41,42,43)(39,47,45,48);(1,5,6,16)(2,10,7,12)(3,13,14,9)(4,8,15,11)(17,21,22,29)(18,24,25,31)(19,27,20,28)(23,30,26,32)(33,34)(35,39)(36,40)(37,38)(41,47)(42,45)(43,48)(44,46)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8808378);