#9683954

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	#44457	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4205 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,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, 9591150);