#9913087

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

4599 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820283);