#7648300

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

4644 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=g,e^2=g,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, 7555496);