#8323158

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

4061 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=h,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8230354);