#8107209

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

4123 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)(3,4)(5,13)(6,7)(8,10)(9,14)(11,16)(12,15)(17,18)(19,28)(20,21)(22,29)(23,27)(24,31)(25,32)(26,30)(34,39)(35,37)(36,44)(47,48);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,19)(18,24)(20,29)(21,30)(22,25)(23,31)(26,32)(27,28)(33,34)(35,40)(36,45)(37,42)(38,39)(41,44)(43,48)(46,47);(2,7)(3,8)(9,13)(11,15)(17,20,23,32)(18,21,27,25)(19,29,31,26)(22,24,30,28)(33,35,38,37)(34,40,39,42)(36,46,44,43)(41,48,45,47);(1,2)(3,10)(4,8)(5,9)(6,7)(11,16)(12,15)(13,14)(17,20)(18,25)(19,29)(21,27)(22,24)(23,32)(26,31)(28,30)(33,36,38,44)(34,41,39,45)(35,43,37,46)(40,47,42,48);(1,4,6,10)(2,8,7,3)(5,12,14,16)(9,15,13,11)(17,21,23,25)(18,20,27,32)(19,30,31,22)(24,29,28,26)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,45)(46,48);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,22,23,30)(18,26,27,29)(19,25,31,21)(20,24,32,28)(33,37,38,35)(34,42,39,40)(36,46,44,43)(41,48,45,47)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014405);