#7644762

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

4339 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)(12,20)(16,23)(18,24)(21,22)(26,29)(27,33)(32,37)(34,36);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,17)(14,19)(18,22)(20,23)(21,24)(25,26,30,29)(27,31,33,35)(28,32,38,37)(34,39,36,40);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,27)(26,31)(28,36)(29,35)(30,33)(32,40)(34,38)(37,39);(9,10)(11,14)(12,16)(13,17)(15,19)(18,21)(20,23)(22,24)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,36)(35,40);(1,4,5,3)(2,8,6,7)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,26)(27,35,33,31)(28,37,38,32)(34,40,36,39)

Polycyclic presentation:

2^9|b^2=h,c^2=gh,f^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, 7551958);