#9812919

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

4665 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=gi,e^2=g,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, 9720115);