#8607445

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	#52031	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4670 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)(3,8)(4,6)(5,7)(10,11)(15,17)(16,18)(22,23)(25,26)(27,32)(28,29)(30,31)(33,38)(34,37)(35,39)(36,40);(2,4)(7,8)(9,10,14,11)(12,17,19,15)(13,16,21,18)(20,23,24,22)(25,26)(27,34)(28,29)(30,31)(32,37)(33,38)(35,36)(39,40);(1,3,6,5)(2,7,4,8)(12,19)(15,17)(20,24)(22,23)(26,31)(28,38)(32,34)(35,40);(1,2,6,4)(3,7,5,8)(9,11,14,10)(12,17,19,15)(13,16,21,18)(20,22,24,23)(25,27)(26,32)(28,35)(29,39)(30,37)(31,34)(33,36)(38,40);(1,4,6,2)(3,7,5,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,28)(26,33)(27,35)(29,31)(30,38)(32,36)(34,39)(37,40);(1,5,6,3)(2,7,4,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,33)(26,28,31,38)(27,36,37,39)(32,40,34,35)

Polycyclic presentation:

2^9|b^2=ghi,c^2=hi,d^2=g,e^2=g,f^2=h,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, 8514641);