#9718533

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

4419 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)(3,6,8,4)(9,10)(11,17)(12,16)(13,18)(14,15)(19,24)(20,21)(22,23)(25,26)(27,31)(28,34)(29,37)(30,32)(33,36)(35,40)(38,39);(1,3)(2,4)(5,8)(6,7)(9,11)(10,13)(12,20)(14,18)(15,17)(16,24)(19,22)(21,23)(25,27)(26,31)(28,35)(29,38)(30,36)(32,33)(34,40)(37,39);(1,2)(3,4)(5,7)(6,8)(10,15)(11,18)(16,23)(19,20)(26,32)(27,36)(34,38)(35,37);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,22)(23,24)(25,27)(26,33)(28,37)(29,38)(30,36)(31,32)(34,40)(35,39);(1,4,5,6)(2,3,7,8)(9,12,14,22)(10,16,15,23)(11,19,18,20)(13,21,17,24)(25,28)(26,34)(27,37)(29,31)(30,39)(32,38)(33,40)(35,36);(1,3)(2,4)(5,8)(6,7)(9,13)(10,11)(12,21)(14,17)(15,18)(16,19)(20,23)(22,24)(25,29,30,40)(26,35,32,37)(27,38,36,34)(28,33,39,31)

Polycyclic presentation:

2^9|a^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9625729);