#10250364

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

4364 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10157560);