#8718969

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

4227 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)(4,12,13,8)(10,15,14,16)(17,18)(19,24)(20,21)(22,27)(23,25)(26,32)(28,31)(29,30)(33,34,38,42)(35,37,41,39)(36,44,46,40)(43,45,48,47);(1,3)(2,7)(4,10)(5,9)(6,11)(8,15)(12,16)(13,14)(17,19,23,27)(18,24,25,22)(20,30,26,28)(21,29,32,31)(33,35,38,41)(34,39,42,37)(36,43,46,48)(40,45,44,47);(4,13)(8,12)(10,14)(15,16)(18,25)(19,27)(20,26)(29,31)(33,35,38,41)(34,39,42,37)(36,43,46,48)(40,45,44,47);(1,4)(2,8)(3,10)(5,12)(6,13)(7,15)(9,16)(11,14)(17,20,23,26)(18,21,25,32)(19,28,27,30)(22,29,24,31)(33,35)(34,39)(36,43)(37,42)(38,41)(40,45)(44,47)(46,48);(1,5,6,2)(3,9,11,7)(4,12,13,8)(10,16,14,15)(17,21)(18,26)(19,29)(20,25)(22,30)(23,32)(24,28)(27,31)(33,36,38,46)(34,40,42,44)(35,43,41,48)(37,47,39,45);(1,3)(2,9)(4,14)(5,7)(6,11)(8,15)(10,13)(12,16)(17,22,23,24)(18,27,25,19)(20,31,26,29)(21,28,32,30)(33,37,38,39)(34,41,42,35)(36,45,46,47)(40,43,44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8626165);