C₂×#10080 #233635

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#10080

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1526 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1526 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);(8,13)(9,14)(10,11)(16,20)(18,25)(21,29)(22,26)(23,27)(28,30);(3,4)(5,6)(7,8)(9,12)(10,14)(11,13)(15,16,19,22)(17,21,24,30)(18,26,25,20)(23,28,27,29);(4,6)(7,9,12,8)(10,14,11,13)(15,17,19,24)(16,21,22,30)(18,27,25,23)(20,28,26,29)

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, 140831);