#8754927

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	#51062	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4347 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)(9,10,15,20)(11,16,23,22)(12,19,14,17)(13,18,21,24)(26,28)(29,37)(32,35)(34,38);(3,7)(5,8)(9,11)(10,16)(12,13)(14,21)(15,23)(17,24)(18,19)(20,22)(25,26,31,28)(27,32,30,35)(29,36,37,33)(34,40,38,39);(1,2)(3,5)(4,6)(7,8)(9,12,15,14)(10,17,20,19)(11,21,23,13)(16,18,22,24)(25,27,31,30)(26,32,28,35)(29,34,37,38)(33,39,36,40);(9,13,15,21)(10,18,20,24)(11,12,23,14)(16,19,22,17)(25,28,31,26)(27,32,30,35)(29,36,37,33)(34,39,38,40);(1,3,4,7)(2,5,6,8)(9,14,15,12)(10,19,20,17)(11,21,23,13)(16,18,22,24)(25,29)(26,33)(27,34)(28,36)(30,38)(31,37)(32,39)(35,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,22)(12,17)(13,18)(14,19)(15,20)(16,23)(21,24)(25,30,31,27)(26,32,28,35)(29,38,37,34)(33,39,36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gi,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662123);