#10368446

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

4270 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10275642);