#9058787

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	#43277	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965983);