#10155822

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

4523 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=ghi,d^2=g,e^2=i,f^2=g,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, 10063018);