#9040517

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	#48363	C₂
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) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 8947713);