#8818144

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

4052 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ghi,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8725340);