#9683922

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	#53862	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4153 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)(18,23)(19,24)(25,26,30,32)(27,28,35,31)(29,33,38,40)(34,36,37,39);(1,2)(3,8)(4,6)(5,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,24)(22,23)(25,27)(26,31)(28,32)(29,37)(30,35)(33,36)(34,38)(39,40);(1,3,5,6)(2,4,7,8)(26,32)(28,31)(33,40)(36,39);(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,21,23,20)(25,28,30,31)(26,27,32,35)(29,36,38,39)(33,34,40,37);(1,2)(3,4)(5,7)(6,8)(9,12,14,16)(10,17,15,11)(13,21,22,20)(18,19,23,24)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,40)(35,37);(1,4)(2,6)(3,7)(5,8)(9,13,14,22)(10,18,15,23)(11,19,17,24)(12,20,16,21)(25,27)(26,31)(28,32)(29,37)(30,35)(33,36)(34,38)(39,40)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=g,e^2=gh,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, 9591118);