#8934820

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
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8842016);