#8599538

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

4567 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gi,d^2=g,e^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,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8506734);