#8334483

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4444 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=ghi,e^2=g,f^2=hi,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, 8241679);