#138027

order	512
nilpotency class	4
isoclinism	isoclinic class of #125098
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

445 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 445 conjugacy classes of subgroups — too many to draw. Only its 98 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)(6,9,10,11)(13,14,18,21,19,22,17,28)(15,23,25,24,26,27,20,16);(5,6,7,11,8,10,12,9)(13,15,24,22,17,25,23,28,19,26,16,14,18,20,27,21);(2,4)(13,16,17,27,19,24,18,23)(14,20,21,15,22,25,28,26)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=g,d^2=ghi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ci,d^b=dh,f^b=fh

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, 45223);