#9712798

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4203 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9619994);