#8202012

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

4528 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)(3,4)(5,6)(7,8)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,26)(27,28)(29,37)(30,32)(31,39)(33,35)(34,40)(36,38);(2,6)(3,8)(10,14)(15,17)(16,21)(22,23)(25,27,30,35)(26,28,32,33)(29,38,39,34)(31,40,37,36);(2,6)(4,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,26,30,32)(27,33,35,28)(29,31,39,37)(34,40,38,36);(1,3)(2,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,38)(39,40);(1,3)(2,7)(4,6)(5,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,29)(26,31)(27,34)(28,36)(30,39)(32,37)(33,40)(35,38);(1,4,5,7)(2,3,6,8)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,26)(27,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)

Polycyclic presentation:

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

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, 8109208);