#8521606

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

3942 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8428802);