#8977167

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
prime radical	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3999 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8884363);