#8922601

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

4530 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(2,5)(3,7)(9,10,13,15)(11,16,19,14)(12,20,21,17)(18,22,24,23)(25,26)(27,36)(28,32)(29,34)(30,35)(31,38)(33,37)(39,40);(1,2,6,5)(3,8,7,4)(9,11,13,19)(10,14,15,16)(12,18,21,24)(17,22,20,23)(25,27,30,38)(26,31,35,36)(28,29,37,39)(32,40,33,34);(1,3)(2,4)(5,8)(6,7)(10,15)(12,21)(14,16)(18,24)(25,28,30,37)(26,32,35,33)(27,29,38,39)(31,40,36,34);(1,2,6,5)(3,4,7,8)(9,10)(11,14)(12,17)(13,15)(16,19)(18,22)(20,21)(23,24)(25,28)(26,33)(27,29)(30,37)(31,34)(32,35)(36,40)(38,39);(1,4,6,8)(2,7,5,3)(9,11)(10,16)(12,18)(13,19)(14,15)(17,23)(20,22)(21,24)(25,29)(26,34)(27,37)(28,38)(30,39)(31,32)(33,36)(35,40);(1,5,6,2)(3,8,7,4)(9,12,13,21)(10,17,15,20)(11,18,19,24)(14,22,16,23)(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=g,c^2=gh,d^2=i,e^2=i,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, 8829797);