#9014327

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	#43103	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4408 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)(3,8)(4,7)(5,6)(9,10,15,18)(11,19,21,16)(12,22,23,13)(14,17,20,24);(3,4)(7,8)(9,11,15,21)(10,16,18,19)(12,17,23,24)(13,20,22,14)(25,26)(27,35)(28,32)(29,33)(30,34)(31,37)(36,40)(38,39);(1,3)(2,7)(4,6)(5,8)(12,23)(13,22)(14,20)(17,24)(25,27,30,37)(26,31,34,35)(28,29,38,36)(32,40,39,33);(3,4)(7,8)(9,12,15,23)(10,13,18,22)(11,17,21,24)(14,16,20,19)(25,28,30,38)(26,32,34,39)(27,36,37,29)(31,33,35,40);(1,4,6,3)(2,7,5,8)(9,13)(10,12)(11,20)(14,21)(15,22)(16,17)(18,23)(19,24)(25,29,30,36)(26,33,34,40)(27,28,37,38)(31,39,35,32);(1,5,6,2)(3,7,4,8)(9,14,15,20)(10,17,18,24)(11,13,21,22)(12,16,23,19)(25,26)(27,35)(28,32)(29,33)(30,34)(31,37)(36,40)(38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=ghi,e^2=i,f^2=gh,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, 8921523);