#10420188

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
prime radical	#47830	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4584 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=h,e^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, 10327384);