#8592013

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	#59074	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4365 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=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, 8499209);