#7901365

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4136 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)(9,10,15,17)(11,19,20,16)(12,22,23,14)(13,24,21,18)(25,26,30,32)(27,33,35,28)(29,37,38,31)(34,36,40,39);(1,3,5,8)(2,6,4,7)(9,11,15,20)(10,16,17,19)(12,21,23,13)(14,18,22,24);(1,2,5,4)(3,6,8,7)(10,17)(11,20)(12,23)(18,24)(25,27,30,35)(26,28,32,33)(29,34,38,40)(31,36,37,39);(1,4,5,2)(3,7,8,6)(9,12)(10,14)(11,21)(13,20)(15,23)(16,18)(17,22)(19,24)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(9,13,15,21)(10,18,17,24)(11,12,20,23)(14,16,22,19)(25,29,30,38)(26,31,32,37)(27,34,35,40)(28,36,33,39);(9,14)(10,12)(11,18)(13,19)(15,22)(16,21)(17,23)(20,24)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7808561);