#9863102

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	#2699	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4092 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^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, 9770298);