#8182936

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

4620 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)(11,17)(12,18)(13,16)(14,15)(19,23)(20,24)(21,22)(25,26)(27,29)(28,31)(30,38)(32,34)(33,40)(35,36)(37,39);(1,2,5,6)(3,7,8,4)(10,15)(11,18)(16,22)(19,24)(25,27,32,31)(26,29,34,28)(30,35,40,39)(33,37,38,36);(2,6)(4,7)(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,24)(22,23)(25,28,32,29)(26,27,34,31)(30,36,40,37)(33,35,38,39);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,24)(22,23)(25,29,32,28)(26,27,34,31)(30,37,40,36)(33,35,38,39);(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,30,32,40)(26,33,34,38)(27,35,31,39)(28,36,29,37);(1,4)(2,3)(5,7)(6,8)(9,13,14,21)(10,16,15,22)(11,19,18,24)(12,20,17,23)(25,31,32,27)(26,28,34,29)(30,39,40,35)(33,36,38,37)

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8090132);