#8932778

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
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4638 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839974);