#8096322

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	#46042	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:

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

Polycyclic presentation:

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

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, 8003518);