#10147825

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

4589 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)(11,14)(12,15)(13,18)(16,21)(17,20)(19,23)(22,24)(25,26,30,32)(27,34,37,31)(28,36,38,33)(29,39,40,35);(2,6)(3,7)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,27,30,37)(26,31,32,34)(28,35,38,39)(29,36,40,33);(1,2,5,6)(3,4,7,8)(10,11)(15,17)(16,18)(22,23)(26,32)(28,38)(31,34)(35,39);(2,6)(3,7)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28,30,38)(26,33,32,36)(27,35,37,39)(29,34,40,31);(1,3)(2,4)(5,7)(6,8)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,27)(26,34)(28,39)(29,33)(30,37)(31,32)(35,38)(36,40);(1,4)(2,7)(3,6)(5,8)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,29)(26,35)(27,36)(28,34)(30,40)(31,38)(32,39)(33,37)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=g,e^2=ghi,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, 10055021);