#8600224

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	#56717	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4357 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:

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

Polycyclic presentation:

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

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, 8507420);