#8341624

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

4449 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)(3,5)(4,7)(6,8)(9,10,15,13)(11,14,17,19)(12,16,21,22)(18,20,23,24)(25,26)(27,32)(28,38)(29,34)(30,35)(31,37)(33,36)(39,40);(2,3)(6,7)(9,11)(10,14)(12,18)(13,19)(15,17)(16,20)(21,23)(22,24)(25,27,30,37)(26,31,35,32)(28,29,36,39)(33,34,38,40);(1,2,5,3)(4,6,8,7)(11,17)(14,19)(18,23)(20,24)(25,27,30,37)(26,32,35,31)(28,29,36,39)(33,40,38,34);(1,3,5,2)(4,7,8,6)(9,12)(10,16)(11,18)(13,22)(14,20)(15,21)(17,23)(19,24)(25,28)(26,33)(27,29)(30,36)(31,34)(32,40)(35,38)(37,39);(1,4)(2,6)(3,7)(5,8)(9,13,15,10)(11,19,17,14)(12,16,21,22)(18,20,23,24)(25,29)(26,34)(27,36)(28,37)(30,39)(31,38)(32,33)(35,40);(9,14,15,19)(10,17,13,11)(12,20,21,24)(16,23,22,18)(25,26)(27,31)(28,33)(29,34)(30,35)(32,37)(36,38)(39,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=h,e^2=hi,f^2=hi,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, 8248820);