#8891662

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4271 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8798858);