#8429428

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4303 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8336624);