#9682732

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4100 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=h,e^2=gh,f^2=g,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, 9589928);