#10261532

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

4437 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=i,e^2=gh,f^2=gi,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, 10168728);