#10142134

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	#59676	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=ghi,e^2=gh,f^2=g,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, 10049330);