#8360916

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

4203 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:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8268112);