#8621969

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	#54839	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4311 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)(3,7)(4,6)(5,8)(9,10,15,18)(11,13,16,21)(12,14,17,23)(19,20,22,24)(25,26,32,38)(27,36,30,33)(28,35,29,34)(31,37,40,39);(1,3,6,8)(2,7,4,5)(9,11,15,16)(10,13,18,21)(12,19,17,22)(14,20,23,24)(25,27)(26,33)(28,37)(29,39)(30,32)(31,35)(34,40)(36,38);(2,4)(3,8)(11,16)(13,21)(19,22)(20,24)(25,28,32,29)(26,34,38,35)(27,37,30,39)(31,33,40,36);(3,8)(5,7)(9,12)(10,14)(11,19)(13,20)(15,17)(16,22)(18,23)(21,24)(25,29,32,28)(26,35,38,34)(27,37,30,39)(31,33,40,36);(1,4,6,2)(3,7,8,5)(9,13,15,21)(10,16,18,11)(12,20,17,24)(14,22,23,19)(25,30,32,27)(26,36,38,33)(28,37,29,39)(31,34,40,35);(1,5,6,7)(2,8,4,3)(9,14)(10,17)(11,20)(12,18)(13,22)(15,23)(16,24)(19,21)(25,31)(26,37)(27,35)(28,33)(29,36)(30,34)(32,40)(38,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^2=g,e^2=gi,f^2=gh,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, 8529165);