#10534325

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

4543 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)(4,6)(5,8)(9,10,14,18)(11,12,16,15)(13,21,22,17)(19,23,24,20)(25,26)(27,30)(28,37)(29,33)(31,35)(32,34)(36,40)(38,39);(1,2,4,6)(3,5,8,7)(25,27)(26,32)(28,36)(29,39)(30,35)(31,34)(33,40)(37,38);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,18)(13,19)(14,16)(17,23)(20,21)(22,24)(25,28,31,29)(26,33,35,37)(27,36,34,39)(30,38,32,40);(1,3,4,8)(2,5,6,7)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,23)(22,24)(25,29,31,28)(26,33,35,37)(27,36,34,39)(30,40,32,38);(9,12)(10,16)(11,18)(13,20)(14,15)(17,24)(19,21)(22,23)(25,27)(26,30)(28,36)(29,39)(31,34)(32,35)(33,38)(37,40);(9,13,14,22)(10,17,18,21)(11,19,16,24)(12,20,15,23)(25,30)(26,34)(27,35)(28,38)(29,40)(31,32)(33,39)(36,37)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,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, 10441521);