#9717473

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	#51891	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4181 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=hi,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, 9624669);