#8097851

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	#2541	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4236 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dhi,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8005047);