#8341906

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	#39405	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4518 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=g,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8249102);