#8928615

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

4554 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)(4,6)(5,8)(9,10,15,17)(11,16,12,22)(13,18,19,14)(20,24,23,21)(25,26)(27,30)(28,31)(29,32)(33,34)(35,37)(36,38)(39,40);(1,2,4,6)(3,5,8,7)(9,11,15,12)(10,16,17,22)(13,20,19,23)(14,21,18,24)(25,26,29,32)(27,30,33,34)(28,31,36,38)(35,37,39,40);(1,2)(3,5)(4,6)(7,8)(10,17)(11,12)(14,18)(20,23)(27,33)(28,36)(30,34)(31,38);(1,3,4,8)(2,5,6,7)(9,12,15,11)(10,16,17,22)(13,20,19,23)(14,24,18,21)(25,27,29,33)(26,30,32,34)(28,35,36,39)(31,37,38,40);(9,13,15,19)(10,18,17,14)(11,20,12,23)(16,24,22,21)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,38)(37,39);(9,14,15,18)(10,19,17,13)(11,21,12,24)(16,23,22,20)(25,28)(26,31)(27,35)(29,36)(30,37)(32,38)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=gh,e^2=h,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, 8835811);