#7651858

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	#42971	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4432 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)(9,10,14,13)(11,17,19,15)(12,16,22,21)(18,24,20,23)(25,26,31,29)(27,35,36,32)(28,33,38,39)(30,34,37,40);(1,3,5,8)(2,6,4,7)(25,27)(26,32)(28,37)(29,35)(30,38)(31,36)(33,34)(39,40);(10,13)(15,17)(16,21)(23,24)(27,36)(30,37)(32,35)(34,40);(1,2,5,4)(3,6,8,7)(9,11)(10,15)(12,20)(13,17)(14,19)(16,24)(18,22)(21,23)(25,28,31,38)(26,33,29,39)(27,37,36,30)(32,34,35,40);(9,12,14,22)(10,16,13,21)(11,18,19,20)(15,23,17,24)(25,29,31,26)(27,35,36,32)(28,33,38,39)(30,40,37,34);(1,4,5,2)(3,7,8,6)(9,13,14,10)(11,17,19,15)(12,21,22,16)(18,24,20,23)(25,30,31,37)(26,34,29,40)(27,38,36,28)(32,33,35,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=g,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,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, 7559054);