#10143442

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3918 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=i,d^2=ghi,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, 10050638);