#10055852

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	#48487	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4163 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)(19,22)(20,23)(21,26)(24,30)(25,27)(28,32)(29,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,47)(44,48);(4,11)(7,9)(12,16)(14,15)(17,18)(19,22)(20,25)(21,24)(23,27)(26,30)(28,32)(29,31)(33,34,37,40)(35,38,42,46)(36,39,44,48)(41,45,43,47);(2,3)(7,9)(8,10)(14,15)(18,19)(20,27)(24,26)(28,29)(33,35,37,42)(34,38,40,46)(36,41,44,43)(39,45,48,47);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,19,22,18)(20,25,27,23)(21,26,30,24)(28,32,29,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,47)(44,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20,22,27)(18,23,19,25)(21,29,30,28)(24,32,26,31)(33,35)(34,38)(36,43)(37,42)(39,47)(40,46)(41,44)(45,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,36)(34,39)(35,41)(37,44)(38,45)(40,48)(42,43)(46,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=h,e^2=i,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, 9963048);