#10408074

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

4698 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,e^2=g,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, 10315270);