#147010

order	512
nilpotency class	4
isoclinism	isoclinic class of #11499
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂⁴	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂²	C₂×#342
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

798 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 798 conjugacy classes of subgroups — too many to draw. Only its 198 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54206);