#9034445

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	#50634	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4430 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8941641);