C₄×#1780 #354776

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

direct product

C₄×#1780

Subgroups

	subgroup	G/H
center	C₂²×C₄²	C₂³
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂⁴	C₂³×C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4170 conjugacy classes of subgroups; only the 706 normal subgroups are shown.

normal subgroups only

This group has 4170 conjugacy classes of subgroups — too many to draw. Only its 706 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) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

(1,2,3,4);(5,6)(7,8)(9,10)(11,12)(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28);(6,8)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(10,12)(15,21)(18,24)(22,27)(25,28)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261972);