#7642554

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	#35051	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4730 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)(12,20)(15,22)(18,24)(21,23)(26,27)(28,35)(32,34)(36,40);(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(25,26,30,27)(28,31,35,33)(29,32,37,34)(36,38,40,39);(1,3,4,7)(2,6,5,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,38,40,39);(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39);(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40)

Polycyclic presentation:

2^9|b^2=hi,c^2=g,e^2=h,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, 7549750);