#8093377

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×#537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4466 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=h,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000573);