#8993465

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

4549 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)(11,19)(12,23)(13,18)(14,15)(16,22)(17,21)(20,24)(25,26,30,32)(27,33,35,29)(28,31,38,36)(34,39,40,37);(1,3,5,7)(2,6,4,8)(9,11,15,22)(10,16,14,19)(12,20,21,13)(17,24,23,18)(25,27)(26,29)(28,34)(30,35)(31,37)(32,33)(36,39)(38,40);(2,4)(3,7)(10,14)(12,21)(13,20)(16,19)(25,26,30,32)(27,29,35,33)(28,36,38,31)(34,39,40,37);(1,3,5,7)(2,6,4,8)(9,12)(10,17)(11,20)(13,22)(14,23)(15,21)(16,24)(18,19)(25,26)(27,29)(28,36)(30,32)(31,38)(33,35)(34,39)(37,40);(1,3)(2,6)(4,8)(5,7)(9,13,15,20)(10,18,14,24)(11,21,22,12)(16,23,19,17)(25,28,30,38)(26,31,32,36)(27,34,35,40)(29,37,33,39);(1,4,5,2)(3,8,7,6)(9,14,15,10)(11,19,22,16)(12,23,21,17)(13,24,20,18)(25,29)(26,27)(28,37)(30,33)(31,34)(32,35)(36,40)(38,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8900661);