#8977165

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	#2424	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)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(17,18,22,26)(19,21,25,23)(20,24,28,32)(27,30,29,31);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,32)(29,30)(33,34,38,39)(35,40,43,48)(36,41,46,45)(37,42,47,44);(2,6)(3,10)(4,13)(14,15)(19,25)(20,28)(21,23)(24,32)(34,39)(35,43)(36,46)(37,47);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,19,22,25)(18,23,26,21)(20,29,28,27)(24,30,32,31)(33,35)(34,40)(36,42)(37,45)(38,43)(39,48)(41,47)(44,46);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,20,22,28)(18,24,26,32)(19,27,25,29)(21,30,23,31)(33,36)(34,41)(35,44)(37,40)(38,46)(39,45)(42,43)(47,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21,22,23)(18,25,26,19)(20,30,28,31)(24,29,32,27)(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=hi,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, 8884361);