#443510

order	512
nilpotency class	3
isoclinism	isoclinic class of #420407
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1284 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1284 conjugacy classes of subgroups — too many to draw. Only its 250 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=g,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^c=hi,f^d=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, 350706);