C₂×#10085 #233631

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#10085

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#2819	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1678 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1678 conjugacy classes of subgroups — too many to draw. Only its 242 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);(8,12)(9,10)(13,14)(15,16,19,22)(17,23,25,30)(18,26,28,21)(20,27,29,24);(3,4)(5,6)(7,8)(9,14)(10,12)(11,13)(15,17)(16,20)(18,27)(19,25)(21,30)(22,29)(23,26)(24,28);(4,6)(7,8)(9,12)(10,14)(11,13)(18,28)(21,26)(23,30)(24,27)

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, 140827);