#8745992

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

4439 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=i,d^2=i,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8653188);