#147085

order	512
nilpotency class	4
isoclinism	isoclinic class of #12019
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#106
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₄×#106
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

548 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)(7,19,20,34)(8,23,24,13)(9,16)(10,28)(11,30)(12,18)(14,22,32,36)(15,26)(17,29)(21,33,25,31)(27,35);(2,4)(5,7,10,21,29,36,27,24,12,20,26,25,11,22,9,8)(6,13,16,14,30,33,15,19,18,23,35,32,17,31,28,34);(5,8,9,22,11,25,26,20,12,24,27,36,29,21,10,7)(6,14,15,23,17,34,16,33,18,32,28,13,30,19,35,31)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,d^2=h,e^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,h^a=hi,f^b=fi,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314209, #72314234, #72314263
maximal normal subgroup of	#72314209, #72314234, #72314263

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54281);