#8914173

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	#51578	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4211 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=h,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

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, 8821369);