#9848736

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

4472 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=ghi,d^2=g,e^2=ghi,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, 9755932);