#9862949

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
prime radical	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4291 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)(10,15)(11,18)(17,20)(19,24)(25,26)(27,35)(28,31)(29,33)(30,39)(32,37)(34,40)(36,38);(1,3,6,5)(2,7,8,4)(9,10)(11,16)(12,20)(13,18)(14,15)(17,22)(19,23)(21,24)(25,27,31,37)(26,32,28,35)(29,30,36,40)(33,34,38,39);(1,4)(2,5)(3,8)(6,7)(26,28)(32,35)(33,38)(34,39);(1,3,6,5)(2,7,8,4)(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,21,20,23)(25,28,31,26)(27,35,37,32)(29,38,36,33)(30,39,40,34);(1,4)(2,3)(5,8)(6,7)(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,23,16,21)(25,29)(26,33)(27,30)(28,38)(31,36)(32,34)(35,39)(37,40);(1,5,6,3)(2,7,8,4)(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,19,20,24)(25,30)(26,34)(27,36)(28,39)(29,37)(31,40)(32,38)(33,35)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=gi,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, 9770145);