#9015546

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	#51768	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4010 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(17,18,23,21)(19,24,22,29)(20,25,27,30)(26,31,28,32)(33,34,39,37)(35,40,45,43)(36,41,47,38)(42,44,46,48);(2,7)(4,15)(8,11)(10,13)(17,19,23,22)(18,24,21,29)(20,26,27,28)(25,31,30,32)(33,35)(34,40)(36,46)(37,43)(38,44)(39,45)(41,48)(42,47);(2,7)(3,12)(4,15)(5,16)(18,21)(19,22)(20,27)(31,32)(33,36,39,47)(34,41,37,38)(35,42,45,46)(40,44,43,48);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,20)(18,25)(19,26)(21,30)(22,28)(23,27)(24,31)(29,32)(33,37,39,34)(35,43,45,40)(36,41,47,38)(42,44,46,48);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,21,23,18)(19,24,22,29)(20,25,27,30)(26,32,28,31)(33,38,39,41)(34,36,37,47)(35,44,45,48)(40,46,43,42);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,22,23,19)(18,24,21,29)(20,28,27,26)(25,31,30,32)(33,35)(34,40)(36,46)(37,43)(38,44)(39,45)(41,48)(42,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=hi,e^2=gi,f^2=h,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, 8922742);