#8969274

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

4477 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,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, 8876470);