#10218528

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

4493 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,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, 10125724);