#7643460

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4334 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7550656);