#9909814

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

4595 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)(3,4,8,6)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,32)(28,37)(29,34)(30,38)(31,33)(35,40)(36,39);(1,3,5,8)(2,6,7,4)(11,17)(15,21)(18,23)(22,24)(25,27)(26,32)(28,29)(30,36)(31,35)(33,40)(34,37)(38,39);(1,3)(2,4)(5,8)(6,7)(10,14)(15,21)(16,19)(22,24)(26,33)(27,35)(28,38)(34,36);(1,4)(2,3)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28)(26,30)(27,29)(31,38)(32,36)(33,37)(34,40)(35,39);(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,29)(26,34)(27,28)(30,40)(31,39)(32,37)(33,36)(35,38);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30)(26,28)(27,36)(29,32)(31,37)(33,38)(34,35)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9817010);