#8093371

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4330 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=gh,e^2=ghi,f^2=g,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, 8000567);