#9040669

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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)(3,10,9,5)(4,8,12,15)(11,16,13,14)(17,18)(19,20)(21,25)(22,30)(23,24)(26,31)(27,32)(28,29);(2,7)(3,9)(8,15)(11,13)(17,19,23,21)(18,20,24,25)(22,27,31,29)(26,28,30,32)(33,34)(35,41)(36,44)(37,38)(39,43)(40,45)(42,47)(46,48);(2,7)(4,12)(5,10)(11,13)(18,24)(19,21)(22,31)(28,32)(34,38)(36,45)(39,41)(42,46);(1,3,6,9)(2,5,7,10)(4,13,12,11)(8,16,15,14)(17,20)(18,19)(21,24)(22,28)(23,25)(26,27)(29,30)(31,32)(33,35)(34,39)(36,46)(37,43)(38,41)(40,48)(42,45)(44,47);(1,4,6,12)(2,8,7,15)(3,11,9,13)(5,14,10,16)(17,21,23,19)(18,25,24,20)(22,29,31,27)(26,32,30,28)(33,36)(34,40)(35,42)(37,45)(38,44)(39,47)(41,48)(43,46);(1,5)(2,9)(3,7)(4,14)(6,10)(8,13)(11,15)(12,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(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=gh,d^2=g,e^2=h,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, 8947865);