#9848496

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
prime radical	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4092 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=g,e^2=hi,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, 9755692);