#9203723

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4480 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9110919);