#8125664

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	#58307	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4222 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8032860);