#7658822

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4246 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7566018);