#8550952

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

4727 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,d^2=h,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,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, 8458148);