#10143383

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	#57200	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4216 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=hi,d^2=i,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, 10050579);