#10419801

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

4575 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gh,e^2=g,f^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, 10326997);