#9716114

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	#56285	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)(9,10,15,19)(11,18,23,16)(12,17,13,14)(20,22,21,24)(25,26)(27,37)(28,29)(30,34)(31,39)(32,36)(33,40)(35,38);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,24)(18,19)(25,27)(26,33)(28,38)(29,39)(30,31)(32,40)(34,35)(36,37);(1,2)(3,4)(5,8)(6,7)(9,12,15,13)(10,17,19,14)(11,20,23,21)(16,24,18,22)(25,28,32,30)(26,34,36,29)(27,38,40,31)(33,35,37,39);(1,4)(2,7)(3,8)(5,6)(9,13,15,12)(10,14,19,17)(11,21,23,20)(16,22,18,24)(25,29,32,34)(26,28,36,30)(27,39,40,35)(31,33,38,37);(1,4)(2,3)(5,6)(7,8)(9,11)(10,18)(12,20)(13,21)(14,24)(15,23)(16,19)(17,22)(25,30,32,28)(26,34,36,29)(27,38,40,31)(33,39,37,35);(1,2)(3,6)(4,7)(5,8)(9,14)(10,12)(11,22)(13,19)(15,17)(16,20)(18,21)(23,24)(25,31,32,38)(26,35,36,39)(27,30,40,28)(29,33,34,37)

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=g,e^2=hi,f^2=hi,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, 9623310);