#6438870

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437608
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	#60368	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3054 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)(5,6,12,17)(7,8,19,23)(9,14,27,33)(10,16,11,15)(13,22,28,21)(18,20,34,35)(24,30,29,36)(25,32,26,31);(1,2,4,3)(5,7,12,19)(6,8,17,23)(9,24,27,29)(10,28,11,13)(14,30,33,36)(15,22,16,21)(18,25,34,26)(20,32,35,31);(1,3,4,2)(5,8)(6,13)(7,16)(9,20)(10,21)(11,22)(12,23)(14,29)(15,19)(17,28)(18,32)(24,33)(25,30)(26,36)(27,35)(31,34);(5,9)(6,14)(7,18)(8,20)(10,25)(11,26)(12,27)(13,29)(15,31)(16,32)(17,33)(19,34)(21,30)(22,36)(23,35)(24,28);(5,10,12,11)(6,15,17,16)(7,13,19,28)(8,21,23,22)(9,25,27,26)(14,31,33,32)(18,29,34,24)(20,30,35,36)

Polycyclic presentation:

2^9|a^2=i,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,e^b=ei,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, 6346066);