#8214225

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
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4199 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=gh,e^2=gi,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, 8121421);