#9863356

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

4494 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=gh,d^2=ghi,e^2=hi,f^2=gh,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, 9770552);