#9938179

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

4376 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=gh,e^2=g,f^2=gh,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, 9845375);