#8907024

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

4543 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,16,19,14)(12,20,21,17)(18,22,24,23)(25,26)(27,32)(28,38)(29,33)(30,34)(31,37)(35,36)(39,40);(1,3,5,8)(2,6,4,7)(9,11,13,19)(10,14,15,16)(12,18,21,24)(17,22,20,23)(25,27)(26,31)(28,35)(29,36)(30,37)(32,34)(33,39)(38,40);(1,3)(2,7)(4,6)(5,8)(10,15)(12,21)(14,16)(18,24)(28,33)(29,38)(35,39)(36,40);(1,3,5,8)(2,6,4,7)(9,10)(11,14)(12,17)(13,15)(16,19)(18,22)(20,21)(23,24)(25,28,30,33)(26,29,34,38)(27,35,37,39)(31,36,32,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,18)(13,19)(14,15)(17,23)(20,22)(21,24)(25,27)(26,32)(28,39)(29,36)(30,37)(31,34)(33,35)(38,40);(1,4,5,2)(3,7,8,6)(9,12,13,21)(10,17,15,20)(11,18,19,24)(14,22,16,23)(25,29)(26,33)(27,36)(28,34)(30,38)(31,39)(32,35)(37,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8814220);