#8944492

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

4478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

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, 8851688);