#9848530

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	#50835	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4341 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)(11,18)(12,20)(13,23)(14,16)(15,21)(17,22)(19,24)(25,26,31,29)(27,35,37,32)(28,38,36,33)(30,34,40,39);(2,3)(6,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,18)(17,24)(22,23)(25,27,31,37)(26,32,29,35)(28,30,36,40)(33,34,38,39);(1,2,5,3)(4,6,8,7)(10,16)(12,22)(15,18)(19,23);(9,12,14,22)(10,17,16,20)(11,19,21,23)(13,15,24,18)(25,28,31,36)(26,33,29,38)(27,30,37,40)(32,34,35,39);(1,3,5,2)(4,6,8,7)(9,11)(10,18)(12,19)(13,17)(14,21)(15,16)(20,24)(22,23)(25,29,31,26)(27,32,37,35)(28,38,36,33)(30,34,40,39);(1,4)(2,6)(3,7)(5,8)(9,13)(10,19)(11,20)(12,15)(14,24)(16,23)(17,21)(18,22)(25,30)(26,34)(27,36)(28,37)(29,39)(31,40)(32,38)(33,35)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=i,d^2=g,e^2=ghi,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, 9755726);