#10143676

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	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4008 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18)(19,24)(20,26)(21,27)(22,30)(23,25)(28,31)(29,32)(33,34,40,44)(35,41,46,48)(36,42,37,39)(38,43,45,47);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,19,23,28)(18,24,25,31)(20,29,30,21)(22,27,26,32)(33,35,40,46)(34,41,44,48)(36,47,37,43)(38,39,45,42);(2,7)(3,10)(8,15)(11,12)(18,25)(19,28)(20,30)(27,32)(33,36,40,37)(34,42,44,39)(35,43,46,47)(38,41,45,48);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,20)(18,26)(19,29)(21,28)(22,25)(23,30)(24,32)(27,31)(33,37,40,36)(34,39,44,42)(35,43,46,47)(38,41,45,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,16)(10,12)(17,21,23,29)(18,27,25,32)(19,30,28,20)(22,31,26,24)(33,38)(34,43)(35,39)(36,41)(37,48)(40,45)(42,46)(44,47);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,22,23,26)(18,20,25,30)(19,27,28,32)(21,24,29,31)(33,39)(34,36)(35,45)(37,44)(38,46)(40,42)(41,47)(43,48)

Polycyclic presentation:

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

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, 10050872);