#8873763

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	#56695	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3960 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8780959);