#9062165

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	#44601	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4059 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)(3,10,11,8)(4,5,9,15)(12,13,14,16)(17,18,23,26)(19,24,27,22)(20,21,25,32)(28,29,30,31)(34,39)(35,44)(37,42)(45,47);(2,7)(3,11)(4,9)(13,16)(17,19,23,27)(18,24,26,22)(20,30,25,28)(21,31,32,29)(33,34)(35,40)(36,42)(37,43)(38,39)(41,44)(45,48)(46,47);(3,11)(4,9)(5,15)(8,10)(19,27)(20,25)(21,32)(22,24)(33,35,38,44)(34,40,39,41)(36,47,43,45)(37,48,42,46);(1,3,6,11)(2,8,7,10)(4,14,9,12)(5,13,15,16)(17,20)(18,21)(19,28)(22,31)(23,25)(24,29)(26,32)(27,30)(33,35)(34,41)(36,47)(37,46)(38,44)(39,40)(42,48)(43,45);(1,4,6,9)(2,5,7,15)(3,12,11,14)(8,16,10,13)(17,21)(18,25)(19,29)(20,26)(22,28)(23,32)(24,30)(27,31)(33,36,38,43)(34,42,39,37)(35,45,44,47)(40,48,41,46);(1,5)(2,9)(3,13)(4,7)(6,15)(8,12)(10,14)(11,16)(17,22)(18,19)(20,31)(21,28)(23,24)(25,29)(26,27)(30,32)(33,37,38,42)(34,43,39,36)(35,46,44,48)(40,47,41,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=gh,e^2=g,f^2=h,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, 8969361);