#10041422

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	#56546	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4134 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)(4,7,11,9)(5,10,13,8)(12,15,16,14)(17,18)(19,21)(20,24)(22,23)(25,30)(26,29)(27,31)(28,32)(33,34,37,38)(35,39,42,41)(36,44,46,40)(43,48,45,47);(4,11)(7,9)(12,16)(14,15)(18,23)(19,25)(20,27)(28,29)(33,34,37,38)(35,41,42,39)(36,40,46,44)(43,48,45,47);(2,3)(7,9)(8,10)(14,15)(18,23)(19,25)(20,27)(28,29)(34,38)(35,42)(40,44)(43,45);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,19,22,25)(18,21,23,30)(20,26,27,32)(24,29,31,28)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,46)(45,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20,22,27)(18,24,23,31)(19,26,25,32)(21,29,30,28)(33,35)(34,39)(36,45)(37,42)(38,41)(40,48)(43,46)(44,47);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,25)(19,23)(20,28)(22,30)(24,26)(27,29)(31,32)(33,36)(34,40)(35,43)(37,46)(38,44)(39,47)(41,48)(42,45)

Polycyclic presentation:

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

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, 9948618);