#8901377

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

4123 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=i,e^2=ghi,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, 8808573);