#9065958

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	#50640	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4415 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=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, 8973154);