#8201996

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

4539 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:

(1,2,4,5)(3,7,8,6)(9,10)(11,18)(12,22)(13,17)(14,15)(16,19)(20,21)(23,24)(25,26)(27,28)(29,37)(30,32)(31,39)(33,35)(34,40)(36,38);(2,5)(6,7)(10,15)(11,19)(12,17)(21,24)(25,27,30,35)(26,28,32,33)(29,38,39,34)(31,40,37,36);(1,2)(3,6)(4,5)(7,8)(10,15)(13,22)(16,18)(21,24)(25,26,30,32)(27,33,35,28)(29,31,39,37)(34,40,38,36);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,20)(13,21)(14,19)(15,18)(17,23)(22,24)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,38)(39,40);(1,3,4,8)(2,6,5,7)(9,12,14,17)(10,13,15,22)(11,20,19,23)(16,21,18,24)(25,29)(26,31)(27,34)(28,36)(30,39)(32,37)(33,40)(35,38);(9,13)(10,17)(11,21)(12,15)(14,22)(16,23)(18,20)(19,24)(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,e^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, 8109192);