#479832

order	512
nilpotency class	3
isoclinism	isoclinic class of #478213
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2842
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

879 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 879 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=eh,c^2=ef,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,g^c=gi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387028);