#8323749

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

4214 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=g,e^2=gh,f^2=gi,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, 8230945);