#7645897

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	#42271	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4305 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)(9,10,13,16)(11,14,19,17)(12,20,21,15)(18,23,24,22)(25,26,31,34)(27,35,38,32)(28,30,37,39)(29,33,36,40);(1,3,5,8)(2,6,4,7)(25,27)(26,32)(28,36)(29,37)(30,33)(31,38)(34,35)(39,40);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(25,28)(26,30)(27,29)(31,37)(32,40)(33,35)(34,39)(36,38);(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24)(25,28)(26,30)(27,36)(29,38)(31,37)(32,33)(34,39)(35,40);(1,4,5,2)(3,7,8,6)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,29,31,36)(26,33,34,40)(27,37,38,28)(30,35,39,32);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,30)(26,28)(27,33)(29,35)(31,39)(32,36)(34,37)(38,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,e^2=g,f^2=h,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, 7553093);