#8531661

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	#35364	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4351 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:

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

Polycyclic presentation:

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

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, 8438857);