#10158582

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

4597 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,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, 10065778);