#9015376

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	#43191	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3917 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)(3,8)(4,10)(5,11)(6,7)(9,13)(12,16)(14,15)(18,20)(19,24)(22,30)(29,32)(33,34,39,37)(35,40,45,43)(36,41,47,38)(42,44,46,48);(1,3,7,15)(2,8,6,14)(4,12,13,5)(9,11,10,16)(17,18,23,20)(19,26,24,21)(22,25,30,28)(27,32,31,29)(33,35)(34,40)(36,46)(37,43)(38,44)(39,45)(41,48)(42,47);(1,4,7,13)(2,9,6,10)(3,5,15,12)(8,16,14,11)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,32)(30,31)(33,36,39,47)(34,41,37,38)(35,42,45,46)(40,44,43,48);(1,4)(2,10)(3,12)(5,15)(6,9)(7,13)(8,16)(11,14)(17,20,23,18)(19,26,24,21)(22,28,30,25)(27,32,31,29)(33,37,39,34)(35,43,45,40)(36,41,47,38)(42,44,46,48);(1,5,7,12)(2,11,6,16)(3,13,15,4)(8,9,14,10)(17,21,23,26)(18,24,20,19)(22,29,30,32)(25,31,28,27)(33,38,39,41)(34,36,37,47)(35,44,45,48)(40,46,43,42);(1,6,7,2)(3,14,15,8)(4,9,13,10)(5,16,12,11)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,35)(34,40)(36,46)(37,43)(38,44)(39,45)(41,48)(42,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=ghi,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8922572);