#7645390

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	#38042	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4397 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)(9,10)(11,15)(12,20)(13,17)(14,19)(16,21)(18,22)(23,24)(25,26,31,33)(27,30,29,39)(28,35,38,32)(34,40,36,37);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,23)(21,24)(25,27,31,29)(26,30,33,39)(28,34,38,36)(32,37,35,40);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,16)(13,17)(14,19)(18,23)(20,21)(22,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,35)(39,40);(9,10)(11,14)(12,16)(13,17)(15,19)(18,22)(20,21)(23,24)(25,29,31,27)(26,30,33,39)(28,36,38,34)(32,37,35,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,22)(21,24)(25,30)(26,29)(27,33)(28,37)(31,39)(32,36)(34,35)(38,40);(1,4,5,3)(2,8,6,7)(9,12,13,21)(10,16,17,20)(11,18,19,24)(14,22,15,23)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,35)(39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=h,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, 7552586);