#9442891

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	#53965	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4064 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9350087);