#10328233

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

4381 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:

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

Polycyclic presentation:

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

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, 10235429);