#8161277

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

4060 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)(8,9)(10,14)(11,15)(12,16)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(34,38)(36,44)(39,41)(42,48);(2,7)(3,9)(10,13)(11,16)(18,19)(23,25)(24,26)(30,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(2,7)(3,9)(10,13)(11,16)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,13)(14,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5,6,14)(2,10,7,13)(3,11,9,16)(4,12,8,15)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|c^2=i,d^2=i,e^2=h,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,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, 8068473);