#8274025

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4361 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

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, 8181221);