#10504947

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4073 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=h,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10412143);