#8602820

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

4592 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)(3,6,8,4)(10,15)(11,17)(16,19)(18,23)(25,26)(27,36)(28,34)(29,33)(30,38)(31,37)(32,39)(35,40);(1,3,5,8)(2,6,7,4)(9,10,14,15)(11,13,17,22)(12,19,20,16)(18,24,23,21)(25,27)(26,32)(28,38)(29,35)(30,33)(31,39)(34,40)(36,37);(2,7)(3,8)(12,20)(16,19)(18,23)(21,24)(25,28,31,29)(26,33,37,34)(27,38,39,35)(30,36,40,32);(1,4,5,6)(2,3,7,8)(9,11)(10,13)(12,18)(14,17)(15,22)(16,21)(19,24)(20,23)(25,29,31,28)(26,34,37,33)(27,38,39,35)(30,36,40,32);(1,2)(3,4)(5,7)(6,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,19)(17,23)(22,24)(25,30)(26,35)(27,34)(28,32)(29,36)(31,40)(33,39)(37,38);(1,4)(2,3)(5,6)(7,8)(9,13)(10,17)(11,15)(12,21)(14,22)(16,23)(18,19)(20,24)(25,27)(26,36)(28,38)(29,35)(30,34)(31,39)(32,37)(33,40)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8510016);