#138704

order	512
nilpotency class	4
isoclinism	isoclinic class of #12483
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#459
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

416 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=i,d^2=ghi,e^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,e^b=eh,f^b=fh,h^b=hi,d^c=dh,e^c=eh,f^c=fh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45900);