#9863040

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	#45094	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4194 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=i,e^2=ghi,f^2=h,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, 9770236);