#10440011

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

4423 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)(3,4,8,6)(9,10)(11,18)(12,15)(13,16)(14,20)(17,22)(19,23)(21,24)(25,26)(27,31)(28,33)(29,39)(30,35)(32,38)(34,36)(37,40);(1,3)(2,6)(4,7)(5,8)(9,11,15,13)(10,16,12,18)(14,19,22,21)(17,23,20,24)(25,27,30,37)(26,31,35,40)(28,36,38,29)(32,39,33,34);(9,10,15,12)(11,18,13,16)(14,17,22,20)(19,24,21,23)(25,28)(26,32)(27,36)(29,37)(30,38)(31,39)(33,35)(34,40);(1,4,5,6)(2,3,7,8)(9,12,15,10)(11,18,13,16)(14,20,22,17)(19,24,21,23)(25,28)(26,33)(27,36)(29,37)(30,38)(31,34)(32,35)(39,40);(1,4)(2,3)(5,6)(7,8)(9,13,15,11)(10,16,12,18)(14,21,22,19)(17,23,20,24)(25,26)(27,31)(28,32)(29,34)(30,35)(33,38)(36,39)(37,40);(1,3)(2,4)(5,8)(6,7)(9,14,15,22)(10,17,12,20)(11,19,13,21)(16,23,18,24)(25,29,30,36)(26,34,35,39)(27,28,37,38)(31,32,40,33)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10347207);