#10534327

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

4743 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)(3,7)(4,5)(6,8)(9,10,14,18)(11,12,16,15)(13,21,22,17)(19,23,24,20)(25,26)(27,28)(29,32)(30,38)(31,34)(33,39)(35,37)(36,40);(2,5)(6,7)(25,27,31,32)(26,29,34,28)(30,35,39,40)(33,37,38,36);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,18)(13,19)(14,16)(17,23)(20,21)(22,24)(25,28,31,29)(26,32,34,27)(30,36,39,37)(33,40,38,35);(1,3,4,8)(2,6,5,7)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,23)(22,24)(25,29,31,28)(26,32,34,27)(30,37,39,36)(33,40,38,35);(9,12)(10,16)(11,18)(13,20)(14,15)(17,24)(19,21)(22,23)(25,26)(27,29)(28,32)(30,38)(31,34)(33,39)(35,36)(37,40);(9,13,14,22)(10,17,18,21)(11,19,16,24)(12,20,15,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,40)(34,38)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gi,d^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10441523);