#10419511

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

4823 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)(3,7)(9,10,14,15)(11,17,12,16)(13,21,22,18)(19,23,20,24)(25,26)(27,29)(28,32)(30,33)(31,34)(35,37)(36,39)(38,40);(2,5)(6,8)(10,15)(16,17)(18,21)(23,24)(25,27,31,28)(26,32,34,29)(30,35,38,36)(33,39,40,37);(1,2)(3,6)(4,5)(7,8)(9,11,14,12)(10,16,15,17)(13,19,22,20)(18,23,21,24)(25,27,31,28)(26,29,34,32)(30,35,38,36)(33,37,40,39);(3,7)(6,8)(9,12,14,11)(10,17,15,16)(13,19,22,20)(18,23,21,24)(25,28,31,27)(26,29,34,32)(30,35,38,36)(33,39,40,37);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,29)(26,28)(27,34)(30,37)(31,32)(33,36)(35,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,13,14,22)(10,18,15,21)(11,19,12,20)(16,23,17,24)(25,30,31,38)(26,33,34,40)(27,35,28,36)(29,37,32,39)

Polycyclic presentation:

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

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, 10326707);