#9675019

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	#50105	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4164 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=h,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, 9582215);