#8549632

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

4384 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8456828);