C₂×#10429 #238559

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#10429

Subgroups

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

1022 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1022 conjugacy classes of subgroups — too many to draw. Only its 226 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) = 512

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

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145755);