#392683

order	512
nilpotency class	3
isoclinism	isoclinic class of #196092
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3700 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^b=fi,e^c=ei,f^c=fi,h^c=hi

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, 299879);