#8362176

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8269372);