#6438855

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437602
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

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

3195 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)(7,12)(8,20)(9,25)(10,14)(11,15)(13,23)(16,33)(17,26)(18,28)(19,29)(21,27)(22,32)(24,31)(30,35)(34,36);(1,2,4,3)(5,7)(6,12)(8,21)(9,18)(10,26)(11,19)(13,30)(14,17)(15,29)(16,22)(20,27)(23,35)(24,34)(25,28)(31,36)(32,33);(1,3,4,2)(5,8)(6,13)(7,16)(9,22)(10,23)(11,24)(12,27)(14,31)(15,32)(17,30)(18,34)(19,35)(20,25)(21,26)(28,33)(29,36);(5,6)(7,17)(8,13)(9,14)(10,25)(11,15)(12,26)(16,30)(18,29)(19,28)(20,23)(21,27)(22,31)(24,32)(33,35)(34,36);(5,9,11,10)(6,14,15,25)(7,18,19,26)(8,22,24,23)(12,17,29,28)(13,31,32,20)(16,34,35,21)(27,30,36,33)

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,e^a=ei,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346051);