#7657218

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4156 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=g,e^2=h,f^2=g,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, 7564414);