#148117

order	512
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#476
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1864
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

720 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fhi,h^b=hi,d^c=di,e^c=ehi,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216650
maximal normal subgroup of	#72216650
maximal quotient of	#72216578

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55313);