#8932879

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×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4860 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^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, 8840075);