#8197139

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	#35129	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4493 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=i,d^2=ghi,e^2=h,f^2=ghi,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, 8104335);