C₄×#2380 #513776

order	512
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

direct product

C₄×#2380

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator	C₂²	C₂³×C₄²
Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4158 conjugacy classes of subgroups; only the 1610 normal subgroups are shown.

normal subgroups only

This group has 4158 conjugacy classes of subgroups — too many to draw. Only its 1610 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 — 5 generators on 24 points:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420972);