#146857

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	C₄×#99
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

666 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,g^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=ci,f^c=fh,f^e=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, 54053);