#8893216

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

4613 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)(11,20)(12,23)(13,14)(15,17)(16,22)(18,21)(19,24)(25,26)(27,29)(28,35)(30,38)(31,33)(32,39)(34,36)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,22)(10,16,17,20)(12,19,21,13)(14,18,24,23)(25,27,31,35)(26,29,33,28)(30,34,39,40)(32,37,38,36);(2,5)(6,8)(10,17)(14,24)(16,20)(18,23)(25,28,31,29)(26,27,33,35)(30,37,39,36)(32,40,38,34);(3,7)(6,8)(9,12)(10,18)(11,19)(13,22)(14,20)(15,21)(16,24)(17,23)(25,29,31,28)(26,27,33,35)(30,37,39,36)(32,34,38,40);(1,3)(2,6)(4,7)(5,8)(9,13,15,19)(10,14,17,24)(11,21,22,12)(16,23,20,18)(25,27)(26,29)(28,33)(30,34)(31,35)(32,37)(36,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,14)(10,19)(11,18)(12,20)(13,17)(15,24)(16,21)(22,23)(25,30,31,39)(26,32,33,38)(27,34,35,40)(28,36,29,37)

Polycyclic presentation:

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

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, 8800412);