#8798702

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

4490 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=ghi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8705898);