#8182855

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

4554 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=g,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8090051);