#8862308

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	#50873	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4279 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:

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

Polycyclic presentation:

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

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, 8769504);