#6438857

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
prime radical	C₂×#3019	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3139 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)(7,11)(8,21)(9,14)(10,15)(12,24)(13,26)(16,30)(17,18)(19,28)(20,29)(22,35)(23,32)(25,31)(27,33)(34,36);(1,2,4,3)(5,7,10,20)(6,11,15,29)(8,22,25,33)(9,18,26,19)(12,30,32,36)(13,28,14,17)(16,23,34,24)(21,35,31,27);(1,3,4,2)(5,8)(6,12)(7,16)(9,23)(10,25)(11,27)(13,31)(14,21)(15,32)(17,30)(18,22)(19,33)(20,34)(24,26)(28,36)(29,35);(5,6)(7,17)(8,12)(9,13)(10,15)(11,18)(14,26)(16,30)(19,29)(20,28)(21,24)(22,27)(23,31)(25,32)(33,35)(34,36);(5,9,10,26)(6,13,15,14)(7,18,20,19)(8,23,25,24)(11,28,29,17)(12,31,32,21)(16,22,34,33)(27,36,35,30)

Polycyclic presentation:

2^9|b^2=fi,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, 6346053);