#6438852

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂×D₄²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3091 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,g^c=gi,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346048);