#8718975

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

4040 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=i,e^2=g,f^2=h,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, 8626171);