#8876042

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

4160 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8783238);