#9938129

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

4371 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9845325);