#9862942

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

4525 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=i,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=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, 9770138);