#7932171

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

4114 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=hi,e^2=ghi,f^2=i,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, 7839367);