#8197799

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

4492 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,e^2=g,f^2=g,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, 8104995);