#8928613

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

4544 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)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,26,31,33)(27,32,28,38)(29,34,35,30)(36,40,39,37);(2,5)(6,7)(10,14)(15,18)(16,21)(22,24)(25,27,31,28)(26,32,33,38)(29,36,35,39)(30,37,34,40);(1,2)(3,6)(4,5)(7,8)(11,17)(12,20)(15,18)(16,21)(26,33)(27,28)(30,34)(36,39);(1,3,4,8)(2,6,5,7)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28,31,27)(26,32,33,38)(29,36,35,39)(30,40,34,37);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,29,31,35)(26,34,33,30)(27,36,28,39)(32,40,38,37);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,30,31,34)(26,35,33,29)(27,37,28,40)(32,39,38,36)

Polycyclic presentation:

2^9|a^2=h,b^2=h,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, 8835809);