#9229603

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	#43816	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4458 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)(11,20)(12,17)(13,22)(14,18)(15,19)(16,23)(21,24)(25,26,29,33)(27,32,35,30)(28,36,38,31)(34,39,37,40);(1,2)(3,4)(5,7)(6,8)(9,11,15,23)(10,16,19,20)(12,21,13,18)(14,22,24,17)(25,26)(27,30)(28,31)(29,33)(32,35)(34,39)(36,38)(37,40);(1,3)(2,6)(4,7)(5,8)(9,12,15,13)(10,17,19,22)(11,21,23,18)(14,20,24,16)(25,27)(26,30)(28,34)(29,35)(31,39)(32,33)(36,40)(37,38);(1,2)(3,6)(4,8)(5,7)(9,13,15,12)(10,17,19,22)(11,18,23,21)(14,20,24,16)(25,28,29,38)(26,31,33,36)(27,34,35,37)(30,39,32,40);(1,4,5,6)(2,3,7,8)(9,10)(11,16)(12,17)(13,22)(14,21)(15,19)(18,24)(20,23)(25,27)(26,32)(28,37)(29,35)(30,33)(31,39)(34,38)(36,40);(1,3)(2,4)(5,8)(6,7)(9,14,15,24)(10,18,19,21)(11,22,23,17)(12,20,13,16)(25,26)(27,30)(28,36)(29,33)(31,38)(32,35)(34,40)(37,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=g,e^2=i,f^2=gh,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, 9136799);