#9909763

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

4510 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,d^2=gi,e^2=ghi,f^2=ghi,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, 9816959);