#10196164

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

4405 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)(9,10,14,15)(11,12,16,20)(13,21,23,17)(18,22,24,19)(25,26)(27,35)(28,33)(29,36)(30,31)(32,37)(34,39)(38,40);(1,3,6,5)(2,7,4,8)(9,11)(10,12)(13,18)(14,16)(15,20)(17,19)(21,22)(23,24)(26,31)(28,38)(32,35)(34,36);(10,15)(11,16)(17,21)(18,24)(25,27,30,37)(26,32,31,35)(28,36,38,34)(29,33,39,40);(2,4)(7,8)(9,11)(10,12)(13,18)(14,16)(15,20)(17,19)(21,22)(23,24)(25,27)(26,32)(28,34)(29,33)(30,37)(31,35)(36,38)(39,40);(1,4,6,2)(3,8,5,7)(9,12)(10,16)(11,15)(13,22)(14,20)(17,18)(19,23)(21,24)(25,28,30,38)(26,33,31,40)(27,36,37,34)(29,32,39,35);(1,5,6,3)(2,7,4,8)(9,13,14,23)(10,17,15,21)(11,18,16,24)(12,19,20,22)(25,29,30,39)(26,34,31,36)(27,33,37,40)(28,35,38,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10103360);