#8760179

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	#44766	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4204 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8667375);