#8402375

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

4537 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)(6,8)(9,10)(11,13)(12,20)(14,16)(15,19)(17,23)(18,24)(21,22)(25,26,31,30)(27,28,32,35)(29,37,40,33)(34,39,38,36);(3,7)(6,8)(9,11)(10,15)(12,21)(13,16)(14,19)(17,22)(18,23)(20,24)(25,27)(26,28)(29,38)(30,35)(31,32)(33,39)(34,40)(36,37);(1,2)(3,6)(4,5)(7,8)(10,16)(13,15)(17,20)(22,24)(26,30)(27,32)(33,37)(34,38);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,21)(13,16)(14,19)(17,22)(18,23)(20,24)(25,28,31,35)(26,32,30,27)(29,39,40,36)(33,34,37,38);(1,2)(3,6)(4,5)(7,8)(9,12,14,23)(10,17,16,20)(11,18,19,21)(13,22,15,24)(25,29)(26,33)(27,34)(28,36)(30,37)(31,40)(32,38)(35,39);(9,13,14,15)(10,11,16,19)(12,22,23,24)(17,18,20,21)(25,30,31,26)(27,35,32,28)(29,37,40,33)(34,39,38,36)

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8309571);