#9779452

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	#45623	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4365 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^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, 9686648);