#148511

order	512
nilpotency class	4
isoclinism	isoclinic class of #13331
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1879
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

682 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:

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

Polycyclic presentation:

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

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	#72216515

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55707);