#8757470

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	#45471	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4330 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664666);