#9064225

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	#43894	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4604 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)(11,19)(12,22)(13,18)(14,16)(15,17)(20,23)(21,24)(25,26,31,36)(27,34,39,32)(28,33,29,38)(30,35,40,37);(1,2)(3,4)(5,6)(7,8)(9,11,15,14)(10,16,17,19)(12,20,18,24)(13,23,22,21)(25,27)(26,32)(28,37)(29,35)(30,38)(31,39)(33,40)(34,36);(2,6)(3,7)(10,17)(13,22)(16,19)(21,23)(25,28,31,29)(26,33,36,38)(27,37,39,35)(30,32,40,34);(3,7)(4,8)(9,12)(10,13)(11,20)(14,24)(15,18)(16,23)(17,22)(19,21)(25,29,31,28)(26,33,36,38)(27,35,39,37)(30,32,40,34);(1,3)(2,4)(5,7)(6,8)(9,13)(10,18)(11,21)(12,17)(14,23)(15,22)(16,20)(19,24)(25,27)(26,34)(28,35)(29,37)(30,38)(31,39)(32,36)(33,40);(1,4)(2,3)(5,8)(6,7)(9,14,15,11)(10,19,17,16)(12,20,18,24)(13,23,22,21)(25,30,31,40)(26,35,36,37)(27,38,39,33)(28,32,29,34)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8971421);