#8202114

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	C₂×#534	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4448 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)(11,15)(12,20)(13,14)(16,21)(17,19)(18,23)(22,24)(25,26)(27,35)(28,37)(29,34)(30,31)(32,36)(33,38)(39,40);(3,7)(5,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,22)(23,24)(26,31)(28,38)(32,35)(34,40);(1,2)(3,5)(4,6)(7,8)(10,14)(15,17)(16,20)(22,23)(25,27,30,36)(26,32,31,35)(28,34,38,40)(29,37,39,33);(1,3,4,7)(2,5,6,8)(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,28)(26,33)(27,34)(29,32)(30,38)(31,37)(35,39)(36,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,28)(26,33)(27,34)(29,32)(30,38)(31,37)(35,39)(36,40);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,29)(26,34)(27,33)(28,32)(30,39)(31,40)(35,38)(36,37)

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8109310);