#8758026

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

4398 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:

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

Polycyclic presentation:

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

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, 8665222);