#8416732

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

3977 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)(3,8)(4,12)(5,10)(6,7)(9,13)(11,16)(14,15)(19,25)(23,26)(27,29)(31,32)(34,36)(39,41)(40,44)(46,47);(2,7)(4,13)(8,16)(10,14)(17,18)(19,26)(20,28)(21,22)(23,25)(24,30)(27,31)(29,32)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(2,7)(5,15)(8,16)(9,12)(18,22)(23,26)(24,28)(31,32)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,3)(2,8)(4,14)(5,9)(6,11)(7,16)(10,13)(12,15)(17,19,21,25)(18,23,22,26)(20,29,30,27)(24,32,28,31)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,4,6,13)(2,9,7,12)(3,10,11,14)(5,8,15,16)(17,20,21,30)(18,24,22,28)(19,27,25,29)(23,31,26,32);(1,5,6,15)(2,10,7,14)(3,9,11,12)(4,8,13,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,30)(29,32)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8323928);