#8141461

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	C₂²×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4596 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)(11,19,12,16)(13,24,17,14)(20,21,22,23)(29,37)(32,39)(35,36)(38,40);(1,3)(2,5)(4,8)(6,7)(11,12)(16,19)(20,22)(21,23)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,38)(37,39);(9,11,15,12)(10,16,18,19)(13,20,17,22)(14,21,24,23)(25,27,30,28)(26,31,33,34)(29,35,37,36)(32,38,39,40);(2,6)(5,7)(9,12,15,11)(10,16,18,19)(13,20,17,22)(14,23,24,21)(25,28,30,27)(26,31,33,34)(29,36,37,35)(32,38,39,40);(1,2)(3,5)(4,6)(7,8)(9,13,15,17)(10,14,18,24)(11,20,12,22)(16,21,19,23)(25,26)(27,31)(28,34)(29,32)(30,33)(35,38)(36,40)(37,39);(1,3)(2,5)(4,8)(6,7)(9,14)(10,17)(11,21)(12,23)(13,18)(15,24)(16,22)(19,20)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8048657);