#8406483

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

4133 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:

(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(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);(2,3)(4,11)(8,10)(12,16)(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);(5,13)(8,10)(12,16)(14,15)(18,21)(23,25)(24,29)(31,32)(34,37)(39,41)(40,44)(47,48);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28)(33,35,38,43)(34,39,37,41)(36,45,46,42)(40,48,44,47);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(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);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(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=gi,b^2=h,d^2=gh,e^2=gh,f^2=h,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, 8313679);