#9015389

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	#51946	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4085 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=ghi,e^2=gi,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, 8922585);