#10409719

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

4723 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10316915);