#9203650

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

4487 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,13)(14,17)(16,18)(19,20)(21,24)(22,23)(25,26,30,31)(27,33,38,32)(28,34,35,29)(36,37,39,40);(2,5)(6,7)(9,11)(10,15)(12,19)(13,20)(14,18)(16,17)(21,22)(23,24)(26,31)(28,35)(32,33)(36,39);(1,2)(3,6)(4,5)(7,8)(11,18)(15,16)(19,21)(20,24)(25,27,30,38)(26,32,31,33)(28,36,35,39)(29,37,34,40);(1,3,4,8)(2,6,5,7)(9,11)(10,16)(12,21)(13,20)(14,18)(15,17)(19,22)(23,24)(25,27)(26,33)(28,39)(29,37)(30,38)(31,32)(34,40)(35,36);(9,12,14,22)(10,13,17,23)(11,19,18,21)(15,20,16,24)(25,28)(26,34)(27,36)(29,31)(30,35)(32,40)(33,37)(38,39);(9,13)(10,12)(11,20)(14,23)(15,19)(16,21)(17,22)(18,24)(25,29,30,34)(26,35,31,28)(27,37,38,40)(32,39,33,36)

Polycyclic presentation:

2^9|a^2=h,c^2=h,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, 9110846);