#147013

order	512
nilpotency class	4
isoclinism	isoclinic class of #11502
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×#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

744 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 194 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)(5,6,9,14,11,16,10,15)(7,17,20,33,18,32,21,34)(8,22,23,31,24,13,25,29)(12,26,28,19,27,36,30,35);(5,7,10,21,11,18,9,20)(6,12,15,28,16,27,14,30)(8,19,23,36,24,35,25,26)(13,17,29,34,22,32,31,33);(1,2,3,4)(7,18)(12,27)(17,32)(19,35)(20,21)(26,36)(28,30)(33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54209);