#7648340

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4538 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:

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

Polycyclic presentation:

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

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, 7555536);