#8720639

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	#59738	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4450 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=i,d^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

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, 8627835);