#8418191

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

4545 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8325387);