#8932877

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

4806 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,e^2=g,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, 8840073);