#10422342

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

4502 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=g,e^2=h,f^2=h,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, 10329538);