#8247578

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4281 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=ghi,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8154774);