#8493228

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

4628 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)(3,8)(4,6)(5,7)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23)(25,26,31,33)(27,34,30,32)(28,38,39,29)(35,40,36,37);(1,3,6,5)(2,7,4,8)(11,12)(13,21)(15,18)(16,23)(25,27,31,30)(26,32,33,34)(28,35,39,36)(29,40,38,37);(9,11,14,12)(10,15,17,18)(13,19,21,20)(16,22,23,24)(25,28)(26,29)(27,35)(30,36)(31,39)(32,40)(33,38)(34,37);(1,2,6,4)(3,7,5,8)(9,12,14,11)(10,15,17,18)(13,20,21,19)(16,22,23,24)(25,28)(26,29)(27,36)(30,35)(31,39)(32,37)(33,38)(34,40);(1,4,6,2)(3,7,5,8)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,29)(26,28)(27,37)(30,40)(31,38)(32,36)(33,39)(34,35);(1,5,6,3)(2,7,4,8)(9,10)(11,15)(12,18)(13,16)(14,17)(19,22)(20,24)(21,23)(25,30,31,27)(26,32,33,34)(28,36,39,35)(29,40,38,37)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=i,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8400424);