#8467220

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	#37348	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4372 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=h,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8374416);