#9059265

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

4111 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)(3,5,10,8)(4,9,12,15)(11,14,13,16)(17,18,23,20)(19,24,21,25)(22,28,30,26)(27,32,29,31);(2,7)(3,10)(4,12)(14,16)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,34)(35,41)(36,44)(37,38)(39,42)(40,45)(43,47)(46,48);(3,10)(4,12)(5,8)(9,15)(18,20)(19,21)(22,30)(31,32)(34,38)(35,42)(36,45)(47,48);(1,3,6,10)(2,8,7,5)(4,13,12,11)(9,14,15,16)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32)(33,35)(34,39)(36,46)(37,42)(38,41)(40,48)(43,45)(44,47);(1,4,6,12)(2,9,7,15)(3,11,10,13)(5,14,8,16)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31)(33,36)(34,40)(35,43)(37,45)(38,44)(39,47)(41,48)(42,46);(1,5,6,8)(2,10,7,3)(4,14,12,16)(9,13,15,11)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,45)(46,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^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, 8966461);