#147378

order	512
nilpotency class	3
isoclinism	isoclinic class of #105681
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1752
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

690 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=ch,f^b=fi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54574);