#8969273

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

4464 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)(11,19)(12,23)(13,18)(14,15)(16,22)(17,21)(20,24)(25,26,30,35)(27,31,29,33)(28,34,38,32)(36,40,37,39);(1,2)(3,4)(5,7)(6,8)(9,11,15,22)(10,16,14,19)(12,20,21,13)(17,24,23,18)(25,27,30,29)(26,31,35,33)(28,37,38,36)(32,40,34,39);(1,3)(2,6)(4,7)(5,8)(10,14)(12,21)(13,20)(16,19)(25,28,30,38)(26,32,35,34)(27,36,29,37)(31,39,33,40);(3,8)(4,6)(9,12)(10,17)(11,20)(13,22)(14,23)(15,21)(16,24)(18,19)(25,29,30,27)(26,33,35,31)(28,36,38,37)(32,39,34,40);(1,4)(2,3)(5,6)(7,8)(9,13,15,20)(10,18,14,24)(11,21,22,12)(16,23,19,17)(25,28)(26,34)(27,36)(29,37)(30,38)(31,40)(32,35)(33,39);(1,2)(3,4)(5,7)(6,8)(9,14,15,10)(11,19,22,16)(12,23,21,17)(13,24,20,18)(25,26)(27,31)(28,34)(29,33)(30,35)(32,38)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8876469);