#8665021

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

4329 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)(11,18)(12,20)(13,15)(14,17)(16,21)(19,23)(22,24)(25,26,30,33)(27,32,34,31)(28,37,38,29)(35,36,40,39);(1,2)(3,8)(4,6)(5,7)(9,11,14,13)(10,15,17,18)(12,19,21,22)(16,23,20,24)(27,34)(31,32)(35,40)(36,39);(1,3,5,6)(2,4,7,8)(25,27)(26,31)(28,35)(29,36)(30,34)(32,33)(37,39)(38,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,16)(11,19)(13,22)(14,21)(15,23)(17,20)(18,24)(25,27)(26,32)(28,35)(29,39)(30,34)(31,33)(36,37)(38,40);(1,2)(3,4)(5,7)(6,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,23)(20,21)(22,24)(25,28,30,38)(26,29,33,37)(27,35,34,40)(31,36,32,39);(1,4)(2,6)(3,7)(5,8)(9,13,14,11)(10,15,17,18)(12,19,21,22)(16,24,20,23)(25,29)(26,28)(27,36)(30,37)(31,35)(32,40)(33,38)(34,39)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8572217);