#7709289

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	#39436	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4488 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)(11,12)(13,17)(14,16)(15,19)(18,20)(21,23)(22,24)(25,26)(27,32)(28,30)(29,37)(31,33)(34,39)(35,40)(36,38);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,27)(26,32)(28,37)(29,30)(31,36)(33,38)(34,35)(39,40);(10,16)(12,15)(17,23)(20,22)(26,33)(27,36)(28,35)(29,39);(9,12,14,15)(10,11,16,19)(13,20,21,22)(17,18,23,24)(25,28,31,35)(26,30,33,40)(27,37,36,34)(29,38,39,32);(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,29)(26,34)(27,30)(28,38)(31,39)(32,35)(33,37)(36,40);(1,3,4,7)(2,6,5,8)(9,13,14,21)(10,17,16,23)(11,18,19,24)(12,20,15,22)(25,30)(26,35)(27,29)(28,33)(31,40)(32,34)(36,39)(37,38)

Polycyclic presentation:

2^9|b^2=ghi,d^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

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, 7616485);