#9779435

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	#50653	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4175 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=gh,e^2=g,f^2=g,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, 9686631);