#8174365

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

4391 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=hi,d^2=gh,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8081561);