#8215213

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4143 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=g,e^2=gi,f^2=gh,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, 8122409);