#512001

order	512
nilpotency class	4
isoclinism	isoclinic class of #512000
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

938 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250086, #72250101, #72250124
maximal normal subgroup of	#72250086, #72250101, #72250124

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419197);