#8932244

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

4530 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)(3,7)(4,5)(6,8)(9,10,15,18)(11,16,19,23)(12,22,21,14)(13,17,20,24)(27,34)(29,37)(31,32)(33,36);(2,5)(6,7)(9,11,15,19)(10,16,18,23)(12,20,21,13)(14,17,22,24)(25,26,30,28)(27,32,34,31)(29,33,37,36)(35,40,38,39);(1,2)(3,6)(4,5)(7,8)(11,19)(12,21)(14,22)(16,23)(25,27,30,34)(26,31,28,32)(29,38,37,35)(33,40,36,39);(1,3,4,8)(2,6,5,7)(9,12)(10,14)(11,20)(13,19)(15,21)(16,17)(18,22)(23,24)(25,27)(26,32)(28,31)(29,35)(30,34)(33,40)(36,39)(37,38);(9,13,15,20)(10,17,18,24)(11,21,19,12)(14,23,22,16)(25,28,30,26)(27,32,34,31)(29,36,37,33)(35,40,38,39);(9,14)(10,12)(11,17)(13,23)(15,22)(16,20)(18,21)(19,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,39)(32,40)(34,38)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=h,d^2=g,e^2=gi,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, 8839440);