#9848383

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

4317 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ghi,d^2=gh,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, 9755579);