#8036990

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×D₄²	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4639 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)(3,7,8,5)(9,10,14,17)(11,13,19,15)(12,16,20,23)(18,21,24,22)(26,31)(27,34)(29,37)(36,38);(1,3,4,8)(2,5,6,7)(12,20)(16,23)(18,24)(21,22)(25,26,30,31)(27,28,34,32)(29,33,37,39)(35,36,40,38);(9,11,14,19)(10,15,17,13)(12,18,20,24)(16,22,23,21)(25,27,30,34)(26,32,31,28)(29,35,37,40)(33,38,39,36);(9,11)(10,13)(12,18)(14,19)(15,17)(16,21)(20,24)(22,23)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,40)(38,39);(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,28)(26,27)(29,38)(30,32)(31,34)(33,40)(35,39)(36,37);(9,13)(10,11)(12,21)(14,15)(16,18)(17,19)(20,22)(23,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,39)(32,38)(34,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7944186);