#10090772

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

4574 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:

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

Polycyclic presentation:

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

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, 9997968);