#348062

order	512
nilpotency class	3
isoclinism	isoclinic class of #348058
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

951 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 951 conjugacy classes of subgroups — too many to draw. Only its 160 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 — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,f^c=fi,g^c=gi,f^d=fi,g^d=gh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255258);