#9836078

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

4466 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9743274);