C₂×#8914 #193019

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#8914

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1234 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 1234 conjugacy classes of subgroups — too many to draw. Only its 298 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) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 30 points:

(1,2);(3,4,6,5)(8,11)(9,13)(12,14)(15,16,20,24)(17,25,27,30)(18,23,19,22)(21,28,29,26);(3,5,6,4)(7,8)(9,14)(10,12)(11,13)(15,17)(16,21)(18,28)(19,26)(20,27)(22,30)(23,25)(24,29);(9,13)(11,14)(15,17)(16,21)(18,26)(19,28)(20,27)(22,25)(23,30)(24,29)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100215);