#8928661

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

4599 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=g,e^2=g,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, 8835857);