#9030091

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	#53432	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4313 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)(4,8)(5,6)(9,10,15,17)(11,16,13,19)(12,18,23,22)(14,21,24,20)(25,26,31,34)(27,32,30,28)(29,33,39,40)(35,37,38,36);(1,2,5,6)(3,7,8,4)(9,11)(10,16)(12,21)(13,15)(14,22)(17,19)(18,24)(20,23)(25,27,31,30)(26,32,34,28)(29,35,39,38)(33,37,40,36);(1,3)(2,4)(5,8)(6,7)(10,17)(11,13)(12,23)(14,24)(25,28,31,32)(26,27,34,30)(29,37,39,36)(33,38,40,35);(1,4)(2,3)(5,7)(6,8)(9,12,15,23)(10,18,17,22)(11,21,13,20)(14,16,24,19)(25,28)(26,27)(29,37)(30,34)(31,32)(33,38)(35,40)(36,39);(1,2)(3,4)(5,6)(7,8)(9,13,15,11)(10,19,17,16)(12,21,23,20)(14,18,24,22)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,37)(34,40);(1,4)(2,3)(5,7)(6,8)(9,14,15,24)(10,20,17,21)(11,22,13,18)(12,16,23,19)(25,30,31,27)(26,28,34,32)(29,38,39,35)(33,36,40,37)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=hi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8937287);