#8188103

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

4468 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095299);