#9040747

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	#50791	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4033 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:

(1,2,6,7)(3,9,10,4)(5,13,14,8)(11,12,16,15)(17,18)(19,24)(20,22)(21,26)(23,25)(27,32)(28,31)(29,30);(2,7)(3,10)(8,13)(11,16)(17,19,23,21)(18,24,25,26)(20,30,27,28)(22,29,32,31)(33,34)(35,41)(36,44)(37,38)(39,43)(40,45)(42,47)(46,48);(2,7)(4,9)(5,14)(11,16)(18,25)(19,21)(22,32)(28,30)(34,38)(36,45)(39,41)(42,46);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(17,20,23,27)(18,22,25,32)(19,28,21,30)(24,31,26,29)(33,35)(34,39)(36,46)(37,43)(38,41)(40,48)(42,45)(44,47);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,21,23,19)(18,26,25,24)(20,28,27,30)(22,31,32,29)(33,36)(34,40)(35,42)(37,45)(38,44)(39,47)(41,48)(43,46);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,22)(18,27)(19,29)(20,25)(21,31)(23,32)(24,28)(26,30)(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=gh,d^2=h,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,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, 8947943);