#7645289

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	#37620	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4358 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:

(2,6)(7,8)(13,21)(16,23)(19,22)(20,24)(26,28)(27,33)(32,36)(35,39);(1,2)(3,7)(4,8)(5,6)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(9,11,14,15)(10,12,17,18)(13,19,21,22)(16,20,23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,3,5,4)(2,7,6,8)(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24)(25,28,30,26)(27,34,33,31)(29,32,37,36)(35,38,39,40);(1,4,5,3)(2,8,6,7)(9,12)(10,15)(11,17)(13,20)(14,18)(16,22)(19,23)(21,24)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40);(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(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,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, 7552485);