#8932213

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

4357 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839409);