#8197209

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

4492 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)(4,6)(5,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(25,26)(27,29)(28,33)(30,31)(32,35)(34,36)(37,38)(39,40);(1,2,4,6)(3,5,8,7)(10,14)(15,21)(16,19)(22,23)(26,31)(29,32)(33,38)(36,39);(1,3)(2,5)(4,8)(6,7)(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,27)(26,32)(28,34)(29,31)(30,35)(33,39)(36,38)(37,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,17)(13,20)(14,19)(15,22)(18,24)(21,23)(25,28,30,37)(26,33,31,38)(27,34,35,40)(29,36,32,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(25,29,30,32)(26,27,31,35)(28,36,37,39)(33,34,38,40)

Polycyclic presentation:

2^9|c^2=g,d^2=gh,e^2=ghi,f^2=ghi,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, 8104405);