#9712835

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4111 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)(3,8)(4,6)(5,7)(9,10,16,20)(11,17,23,24)(12,13,14,18)(15,22,21,19)(25,26,31,35)(27,34,30,32)(28,29,33,40)(36,37,38,39);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,22)(13,21)(14,19)(15,18)(16,23)(20,24)(25,27,31,30)(26,32,35,34)(28,36,33,38)(29,39,40,37);(9,12,16,14)(10,18,20,13)(11,19,23,22)(15,17,21,24)(25,27,31,30)(26,32,35,34)(28,38,33,36)(29,37,40,39);(9,13,16,18)(10,12,20,14)(11,21,23,15)(17,22,24,19)(25,28,31,33)(26,29,35,40)(27,36,30,38)(32,39,34,37);(1,2)(3,6)(4,8)(5,7)(9,14,16,12)(10,18,20,13)(11,22,23,19)(15,17,21,24)(25,29)(26,33)(27,37)(28,35)(30,39)(31,40)(32,36)(34,38);(1,4,5,3)(2,6,7,8)(9,15,16,21)(10,19,20,22)(11,18,23,13)(12,17,14,24)(25,30,31,27)(26,34,35,32)(28,38,33,36)(29,37,40,39)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=h,e^2=i,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, 9620031);