#8174284

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

4571 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)(25,26)(27,36)(28,29)(30,39)(31,35)(32,38)(33,37)(34,40);(1,3,6,4)(2,7,8,5)(10,15)(11,18)(17,20)(19,24)(25,27,31,38)(26,32,35,36)(28,34,33,39)(29,30,37,40);(1,3,6,4)(2,7,8,5)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,28,31,33)(26,29,35,37)(27,34,38,39)(30,36,40,32);(2,8)(3,4)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,29,31,37)(26,33,35,28)(27,30,38,40)(32,39,36,34);(1,4,6,3)(2,5,8,7)(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,26)(27,32)(28,29)(30,34)(31,35)(33,37)(36,38)(39,40);(1,5)(2,3)(4,8)(6,7)(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,19,20,24)(25,30)(26,34)(27,37)(28,36)(29,38)(31,40)(32,33)(35,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gi,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, 8081480);