#9724824

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	#40372	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4243 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9632020);