#7642588

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	#38035	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4407 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549784);