#8418182

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

4283 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=h,e^2=gh,f^2=gi,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, 8325378);