#512113

order	512
nilpotency class	4
isoclinism	isoclinic class of #511893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

958 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)(5,6,9,15)(7,19,25,34)(8,26,23,36)(10,17,28,30)(11,16,21,29)(12,32,20,18)(13,31,33,22)(14,24,35,27);(5,6)(7,13)(8,14)(9,15)(10,29)(11,30)(12,18)(16,28)(17,21)(19,31)(20,32)(22,34)(23,35)(24,26)(25,33)(27,36);(2,4)(5,7,20,24,9,25,12,27)(6,13,32,26,15,33,18,36)(8,21,22,28,23,11,31,10)(14,17,34,16,35,30,19,29);(5,8)(6,14)(7,21)(9,23)(10,27)(11,25)(12,31)(13,17)(15,35)(16,26)(18,19)(20,22)(24,28)(29,36)(30,33)(32,34)

Polycyclic presentation:

2^9|a^2=i,c^2=eh,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,h^a=hi,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,g^c=gh,h^c=hi,e^d=ei,f^d=fh,g^d=ghi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250062, #72250074, #72250086
maximal normal subgroup of	#72250062, #72250074, #72250086

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419309);