#8188173

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	#46368	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4473 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)(3,7)(4,6)(5,8)(9,10,14,13)(11,17,19,15)(12,20,21,16)(18,22,24,23)(25,26,32,34)(27,29,31,28)(30,33,39,40)(35,37,38,36);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,27,32,31)(26,28,34,29)(30,35,39,38)(33,36,40,37);(1,2,4,6)(3,5,8,7)(10,13)(15,17)(16,20)(22,23)(25,28,32,29)(26,27,34,31)(30,36,39,37)(33,35,40,38);(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,29,32,28)(26,27,34,31)(30,37,39,36)(33,35,40,38);(1,3)(2,5)(4,8)(6,7)(9,12,14,21)(10,16,13,20)(11,18,19,24)(15,22,17,23)(25,30,32,39)(26,33,34,40)(27,35,31,38)(28,36,29,37);(1,2)(3,5)(4,6)(7,8)(9,13,14,10)(11,17,19,15)(12,20,21,16)(18,23,24,22)(25,31,32,27)(26,28,34,29)(30,38,39,35)(33,36,40,37)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095369);