#10183062

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

4427 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=i,d^2=ghi,e^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, 10090258);