#98527

order	512
nilpotency class	3
isoclinism	isoclinic class of #97372
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

1192 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 1192 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,f^b=fgh,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5723);