#10444936

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

4677 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:

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

Polycyclic presentation:

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

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, 10352132);