#9863075

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	C₂×#562	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4086 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=i,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9770271);