#8094728

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

4446 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8001924);