#8969254

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
prime radical	#54389	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4302 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gh,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, 8876450);