#8904778

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	#45439	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4878 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,e^2=g,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, 8811974);