#9229474

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	#43871	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4317 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9136670);