#8873740

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3886 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=hi,d^2=i,e^2=gi,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, 8780936);