#8922636

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

4512 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=gi,e^2=i,f^2=gh,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, 8829832);