#99993

order	512
nilpotency class	3
isoclinism	isoclinic class of #99979
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#459	C₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

572 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 256

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

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

Polycyclic presentation:

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

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, 7189);