#9815348

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

4460 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=ghi,e^2=g,f^2=gh,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, 9722544);