#9863872

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

4606 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9771068);