C₂×#10428 #238558

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

Aliases

direct product

C₂×#10428

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

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

normal subgroups only

This group has 978 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) = 38
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 38 points:

(1,2);(4,5)(7,8)(9,13)(10,14)(11,15)(12,16)(17,20)(18,21)(19,22)(23,24,27,31)(25,28,33,38)(26,29,35,30)(32,36,34,37);(3,4)(5,6)(7,9,12,19)(8,13,16,22)(10,18,11,17)(14,21,15,20)(23,25,27,33)(24,28,31,38)(26,34,35,32)(29,37,30,36);(3,5,6,4)(8,14)(9,17)(10,11)(15,16)(18,19)(20,21)(24,29)(26,35)(28,36)(30,31)(32,34)(37,38)

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, 145754);