#7657123

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	#34597	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4245 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)(11,21)(12,17)(13,18)(14,19)(15,20)(16,24)(22,23)(26,31)(27,35)(33,34)(36,37);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,22)(13,19)(14,23)(15,24)(17,18)(20,21)(26,31)(32,38)(33,34)(39,40);(9,11)(10,16)(12,22)(13,19)(14,23)(15,24)(17,18)(20,21)(25,26,30,31)(27,32,35,38)(28,33,29,34)(36,39,37,40);(1,3,4,7)(2,6,5,8)(9,12,15,19)(10,17,20,14)(11,22,24,13)(16,18,21,23)(25,27)(26,32)(28,36)(29,37)(30,35)(31,38)(33,39)(34,40);(9,13,15,22)(10,18,20,23)(11,19,24,12)(14,16,17,21)(25,28,30,29)(26,33,31,34)(27,36,35,37)(32,39,38,40);(9,14,15,17)(10,19,20,12)(11,23,24,18)(13,21,22,16)(25,29,30,28)(26,34,31,33)(27,37,35,36)(32,40,38,39)

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7564319);