#10363660

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

4508 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)(9,10,14,12)(11,17,19,15)(13,20,22,16)(18,23,21,24)(25,26,31,34)(27,32,28,30)(29,33,38,40)(35,39,36,37);(9,11,14,19)(10,15,12,17)(13,21,22,18)(16,24,20,23)(27,28)(30,32)(35,36)(37,39);(1,2)(3,6)(4,5)(7,8)(10,12)(15,17)(16,20)(23,24)(25,27,31,28)(26,32,34,30)(29,36,38,35)(33,37,40,39);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,23)(20,24)(21,22)(25,28,31,27)(26,30,34,32)(29,35,38,36)(33,39,40,37);(1,3)(2,6)(4,7)(5,8)(9,12,14,10)(11,17,19,15)(13,20,22,16)(18,24,21,23)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,22)(10,16,12,20)(11,18,19,21)(15,23,17,24)(25,30)(26,27)(28,34)(29,37)(31,32)(33,35)(36,40)(38,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10270856);